<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls7 ws4">Sistemas de Equações</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls8 ws4">Lineares e Matrizes</div><div class="t m0 x2 h2 y3 ff1 fs0 fc1 sc0 ls7 ws4">Sistemas de Equações</div><div class="t m0 x2 h2 y4 ff1 fs0 fc1 sc0 ls8 ws4">Lineares e Matrizes</div><div class="t m0 x3 h3 y5 ff1 fs1 fc2 sc0 ls9 ws5">Conteúdo do Capítulo</div><div class="t m0 x4 h4 y6 ff2 fs2 fc1 sc0 lsa ws0">1.1 <span class="ff3 fc2 ws6">Introdução aos Sistemas de Equações Lineares</span></div><div class="t m0 x4 h4 y7 ff2 fs2 fc1 sc0 lsa ws0">1.2 <span class="ff3 fc2 ws6">Eliminação Gaussiana</span></div><div class="t m0 x4 h4 y8 ff2 fs2 fc1 sc0 lsa ws0">1.3 <span class="ff3 fc2 ws6">Matrizes e Operações Matriciais</span></div><div class="t m0 x4 h4 y9 ff2 fs2 fc1 sc0 lsa ws0">1.4 <span class="ff3 fc2 ws6">Inversas; Regras da <span class="_0 blank"></span>Aritmética Matricial</span></div><div class="t m0 x4 h5 ya ff2 fs2 fc1 sc0 lsa ws0">1.5 <span class="ff3 fc2 ws6">Matrizes Elementares e um Método para Encontrar <span class="ff4 ls0">A</span><span class="fs3 lsb ws8 v1">\u2013 1</span></span></div><div class="t m0 x4 h4 yb ff2 fs2 fc1 sc0 lsa ws0">1.6 <span class="ff3 fc2 ws6">Mais Resultados sobre Sistemas de Equações e Invertibilidade</span></div><div class="t m0 x4 h4 yc ff2 fs2 fc1 sc0 lsa ws0">1.7 <span class="ff3 fc2 ws6">Matrizes Diagonais, T<span class="_0 blank"></span>riangulares e Simétricas</span></div><div class="t m0 x5 h6 yd ff5 fs4 fc1 sc0 lsb ws1">I<span class="fs2 lsc ws2 v2">NTRODUÇÃO: <span class="fc2 lsa ws9">Muitas vezes na Ciência e na Matemática a informação é organizada em linhas e</span></span></div><div class="t m0 x6 h7 ye ff5 fs2 fc2 sc0 lsa wsa">colunas formando agrupamentos retangulares chamados matrizes. Estas matrizes podem ser tabelas</div><div class="t m0 x6 h7 yd ff5 fs2 fc2 sc0 lsa wsb">de dados numéricos surgidos de observações físicas, mas também ocorrem em vários contextos</div><div class="t m0 x5 h7 yf ff5 fs2 fc2 sc0 lsa wsc">matemáticos. Por exemplo, nós veremos neste capítulo que para resolver um sistema de equações tal</div><div class="t m0 x5 h7 y10 ff5 fs2 fc2 sc0 lsa">como</div><div class="t m0 x5 h7 y11 ff5 fs2 fc2 sc0 lsa wsd">toda a informação requerida para chegar à solução está encorpada na matriz</div><div class="t m0 x5 h7 y12 ff5 fs2 fc2 sc0 lsa wse">e que a solução pode ser obtida efetuando operações apropriadas nesta matriz. Isto é particularmente</div><div class="t m0 x5 h7 y13 ff5 fs2 fc2 sc0 lsa wsf">importante no desenvolvimento de programas de computador para resolver sistemas de equações li-</div><div class="t m0 x5 h7 y14 ff5 fs2 fc2 sc0 lsa ws10">neares, porque os computadores são muito bons para manipular coleções de números. Contudo, as</div><div class="t m0 x5 h7 y15 ff5 fs2 fc2 sc0 lsa ws11">matrizes não são simplesmente uma ferramenta de notação para resolver sistemas de equações linea-</div><div class="t m0 x5 h7 y16 ff5 fs2 fc2 sc0 lsa ws12">res; elas também podem ser vistas como objetos matemáticos de vida própria, existindo uma teoria rica</div><div class="t m0 x5 h7 y17 ff5 fs2 fc2 sc0 lsa ws13">e importante a elas associada que tem uma grande variedade de aplicações. Neste capítulo nós iremos</div><div class="t m0 x5 h7 y18 ff5 fs2 fc2 sc0 lsa wsd">começar o estudo de matrizes.</div><div class="c x7 y19 w2 h8"><div class="t m1 x0 h9 y1a ff6 fs5 fc3 sc0 ls1">\ue002<span class="ff7 lsb v3">5</span></div><div class="t m1 x8 ha y1b ff7 fs5 fc3 sc0 lsb">2</div><div class="t m1 x9 ha y1c ff7 fs5 fc3 sc0 lsb">1</div><div class="t m1 xa ha y1d ff8 fs5 fc3 sc0 lsb">\u2212<span class="ff7">1</span></div><div class="t m1 xb ha y1e ff7 fs5 fc3 sc0 lsb">3</div><div class="t m1 xb hb y1f ff7 fs5 fc3 sc0 ls2">4<span class="ff6 lsb v4">\ue003</span></div></div><div class="c xc y20 w3 hc"><div class="t m1 xd ha y21 ff9 fs5 fc3 sc0 lsb">5<span class="ffa ls3">x<span class="ffb ls4">+</span><span class="ls5">y<span class="ffb ls6">=</span></span></span>3</div><div class="t m1 xd ha y22 ff9 fs5 fc3 sc0 ls1">2<span class="ffa ls3">x<span class="ffb ls4">\u2212</span><span class="ls5">y<span class="ffb ls6">=</span></span></span><span class="lsb">4</span></div></div><div class="t m0 xe hd y23 ff1 fs6 fc2 sc0 lsb ws3">1<span class="_1 blank"></span><span class="fc4 sc1">1</span></div><div class="t m0 xf he y24 ff1 fs7 fc2 sc0 lsd">27</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi xb y25 w4 hf" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg2.png"><div class="t m0 x1 h10 y26 ff1 fs8 fc1 sc0 ls6f ws14">1.1 <span class="fs9 fc2 ls70 ws2f v5">INTRODUÇÃO AOS SISTEMAS DE</span></div><div class="t m0 x10 h11 y27 ff1 fs9 fc2 sc0 ls70 ws2f">EQUAÇÕES LINEARES</div><div class="t m0 xb h7 y28 ffc fs2 fc5 sc0 lsa ws30">Os sistemas de equações algébricas lineares e suas soluções</div><div class="t m0 xb h7 y29 ffc fs2 fc5 sc0 lsa ws31">constituem um dos principais tópicos estudados em cursos co-</div><div class="t m0 xb h7 y2a ffc fs2 fc5 sc0 lsa ws32">nhecidos como \u201cde Álgebra Linear.\u201d Nesta primeira seção nós</div><div class="t m0 xb h7 y2b ffc fs2 fc5 sc0 lsa ws33">iremos introduzir alguma terminologia básica e discutir um méto-</div><div class="t m0 xb h7 y2c ffc fs2 fc5 sc0 lsc ws34">do para resolver estes sistemas.</div><div class="t m0 xb he y2d ff1 fs7 fc1 sc0 ls71 ws35">Equações Lineares<span class="_2 blank"> </span><span class="ff3 fs2 fc2 lsb ws36">Qualquer linha reta no plano <span class="ff4 lsa ws15">xy </span>pode</span></div><div class="t m0 xb h4 y2e ff3 fs2 fc2 sc0 lsa ws6">ser representada algebricamente por uma equação da forma</div><div class="t m0 xb h12 y2f ff3 fs2 fc2 sc0 lsb ws7">onde <span class="_0 blank"></span><span class="ff4 lse">a<span class="ff3 fs3 lsb ws16 v6">1</span><span class="ff3 lsf v0">,<span class="ff4 lsb ws17">a</span><span class="fs3 lsb v6">2 </span></span></span></div><div class="t m0 x11 h4 y30 ff3 fs2 fc2 sc0 lsb ws7">e <span class="_0 blank"></span><span class="ff4 ls10">b<span class="ff3 lsa ws37">são constantes reais e </span><span class="ls11">a<span class="ff3 fs3 ls12 v6">1</span></span><span class="ff3">e</span><span class="ls13">a<span class="ff3 fs3 lsb v6">2 </span></span></span></div><div class="t m0 x12 h4 y30 ff3 fs2 fc2 sc0 lsa ws37">não são ambas nulas.</div><div class="t m0 xb h4 y31 ff3 fs2 fc2 sc0 lsb ws38">Uma equação desta forma é chamada uma equação linear nas</div><div class="t m0 xb h4 y32 ff3 fs2 fc2 sc0 lsa ws7">variáveis <span class="_3 blank"> </span><span class="ff4 ls14">x<span class="ff3">e</span><span class="lsb">y</span></span><span class="ws39">. Mais geralmente, nós definimos uma <span class="ffd">equação</span></span></div><div class="t m0 xb h4 y33 ffd fs2 fc2 sc0 lsa ws18">linear <span class="ff3 ws7">nas <span class="ff4 ls15">n</span><span class="ws19">variáveis <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">1</span></span><span class="ls15">,<span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws3a">,. . ., </span><span class="ls16">x<span class="fs3 ls17 v6">n</span></span><span class="ff3 ws3a">como uma equação que pode</span></span></span></span></span></div><div class="t m0 xb h4 y34 ff3 fs2 fc2 sc0 lsa ws6">ser expressa na forma</div><div class="t m0 xb h13 y35 ff3 fs2 fc2 sc0 lsb ws7">onde <span class="_4 blank"> </span><span class="ff4 ls18">a</span><span class="fs3 ws16 v6">1</span><span class="ls19 v0">,<span class="ff4 ls1a">a</span></span><span class="fs3 ws16 v6">2</span><span class="ws3b v0">, ..., <span class="ff4 ws17">a<span class="fs3 ls1b v6">n</span></span><span class="ws7">e <span class="_4 blank"> </span><span class="ff4 ls19">b</span></span><span class="lsa">são constantes reais. <span class="_0 blank"></span>As variáveis de uma</span></span></div><div class="t m0 xb h4 y36 ff3 fs2 fc2 sc0 lsb ws3c">equação linear são, muitas vezes, chamadas <span class="ffd lsa">incógnitas</span>.</div><div class="t m0 xb h4 y37 ff3 fs2 fc2 sc0 lsa ws6">As equações</div><div class="t m0 xb h4 y38 ff3 fs2 fc2 sc0 lsb ws3d">são lineares. Observe que uma equação linear não envolve</div><div class="t m0 xb h4 y39 ff3 fs2 fc2 sc0 lsa ws3e">quaisquer produtos ou raízes de variáveis. T<span class="_5 blank"></span>odas as variáveis</div><div class="t m0 xb h4 y3a ff3 fs2 fc2 sc0 lsb ws3f">ocorrem somente na primeira potência e não aparecem como</div><div class="t m0 xb h4 y3b ff3 fs2 fc2 sc0 lsa ws40">argumentos de funções trigonométricas, logarítmicas ou expo-</div><div class="t m0 xb h4 y3c ff3 fs2 fc2 sc0 lsa ws41">nenciais. As <span class="_4 blank"> </span>equações</div><div class="t m0 xb h14 y3d ff3 fs2 fc2 sc0 lsa ws7">são <span class="_4 blank"> </span><span class="ff4">não</span><span class="ws1a">-lineares. <span class="ffe fc1 lsb">®</span></span></div><div class="t m0 x13 h4 y3e ff3 fs2 fc2 sc0 lsa ws7">Uma <span class="ffd ws1b">solução </span><span class="lsb ws42">de uma equação linear <span class="ff4 ws17">a</span><span class="fs3 ws7 v6">1 </span><span class="ff4 ws17 v0">x</span><span class="fs3 ls1c v6">1</span></span><span class="v0">+ <span class="_4 blank"> </span><span class="ff4 ls1d">a</span><span class="fs3 ls1e v6">2</span><span class="ff4 lsb ws17">x</span><span class="fs3 ls1c v6">2</span><span class="lsb ws42">+ · · · + <span class="ff4 ls1f">a<span class="fs3 lsb v6">n</span></span></span></span></div><div class="t m0 xb h4 y3f ff4 fs2 fc2 sc0 lsb ws17">x<span class="fs3 ws7 v6">n <span class="_3 blank"> </span></span><span class="ff3 lsa ws7">= <span class="_6 blank"> </span></span><span class="ls20">b</span><span class="ff3 ws43">é uma seqüência de </span><span class="ls20">n<span class="ff3 lsa ws7">números <span class="_6 blank"> </span></span></span>s<span class="ff3 fs3 ws16 v6">1</span><span class="ff3 ls20">,</span>s<span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws44">, ..., </span><span class="ls21">s<span class="fs3 ls22 v6">n</span></span><span class="ff3 ws43">tais que a</span></div><div class="t m0 xb h4 y40 ff3 fs2 fc2 sc0 lsa ws45">equação é satisfeita quando substituímos <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws7 v6">1 </span></span><span class="ws7">= <span class="ff4 lsb ws17">s<span class="ff3 fs3 ws16 v6">1</span></span><span class="ls23">,<span class="ff4 lsb ws17">x</span><span class="fs3 ls24 v6">2</span></span>= <span class="_4 blank"> </span><span class="ff4 lsb ws17">s<span class="ff3 fs3 ws16 v6">2</span></span></span><span class="lsb">, ..., <span class="ff4 ws17">x<span class="fs3 ls24 v6">n</span></span>=</span></div><div class="t m0 xb h4 y41 ff4 fs2 fc2 sc0 ls25">s<span class="fs3 lsb ws16 v6">n</span><span class="ff3 lsb ws46">. O conjunto de todas as soluções de uma equação é chamado</span></div><div class="t m0 xb h4 y42 ff3 fs2 fc2 sc0 lsa ws7">seu <span class="ffd ws1c">conjunto-solução </span><span class="ls26 ws47">ou, às vezes, a </span><span class="ffd ws48">solução geral <span class="ff3 lsb">da equação.</span></span></div><div class="t m0 xb h4 y43 ff3 fs2 fc2 sc0 lsb ws49">Encontre o conjunto-solução de (a) 4<span class="ff4 ws7">x <span class="_4 blank"> </span></span>\u2013 2<span class="ff4 ws7">y <span class="_4 blank"> </span></span><span class="lsa ws4a">= 1 e (b) <span class="ff4 ls27">x</span><span class="fs3 ls28 v6">1</span></span><span class="v0">\u2013 4<span class="ff4 ws17">x</span><span class="fs3 ls28 v6">2</span>+</span></div><div class="t m0 xb h4 y44 ff3 fs2 fc2 sc0 lsb">7<span class="ff4 ws17">x</span><span class="fs3 ls29 v6">3</span><span class="ws6">= 5.</span></div><div class="t m0 xb h4 y45 ff4 fs2 fc1 sc0 lsb ws7">Solução <span class="_2 blank"> </span><span class="ff3">(</span>a<span class="ff3 ls72 ws1d">). <span class="fc2 lsa ws4b">Para encontrar soluções de (a), nós podemos</span></span></div><div class="t m0 xb h4 y46 ff3 fs2 fc2 sc0 lsa ws4c">atribuir um valor arbitrário a <span class="ff4 ls2a">x</span>e resolver em <span class="ff4 lsb">y<span class="ff3 ws4a">, ou escolher um</span></span></div><div class="t m0 xb h4 y47 ff3 fs2 fc2 sc0 lsa ws4d">valor arbitrário para <span class="ff4 ls2b">y</span>e resolver em <span class="ff4 lsb">x</span>. Seguindo a primeira</div><div class="t m0 xb h4 y48 ff3 fs2 fc2 sc0 lsb ws3c">abordagem e dando um valor arbitrário <span class="ff4 ls2c">t</span><span class="lsa ws1e">para </span><span class="ff4">x</span>, obtemos</div><div class="t m0 xb h4 y49 ff3 fs2 fc2 sc0 lsb ws4e">Estas fórmulas descrevem o conjunto-solução em termos de um</div><div class="t m0 xb h4 y4a ff3 fs2 fc2 sc0 lsa ws4f">número arbitrário <span class="ff4 lsb">t<span class="ff3">, chamado </span><span class="lsa ws1f">parâmetr<span class="_0 blank"></span>o<span class="ff3 ws4f">. Soluções numéricas</span></span></span></div><div class="t m0 x14 h4 y4b ff3 fs2 fc2 sc0 lsa ws50">particulares podem ser obtidas substituindo <span class="ff4 ls2d">t</span>por valores especí-</div><div class="t m0 x14 h4 y4c ff3 fs2 fc2 sc0 lsa ws51">ficos. Por exemplo, <span class="ff4 ls2e">t</span><span class="lsb">= 3 dá a solução <span class="ff4">x = </span><span class="ws7">3, <span class="ff4">y </span></span></span><span class="ws52">= e <span class="_7 blank"></span><span class="ff4 ls2e">t<span class="ff3 lsa ws53">= dá</span></span></span></div><div class="t m0 x14 h4 y4d ff3 fs2 fc2 sc0 lsb ws3c">a solução <span class="ff4 ws7">x </span><span class="lsa ws54">= , <span class="_8 blank"></span><span class="ff4 lsb ws7">y <span class="ff3 lsa ws55">= .</span></span></span></div><div class="t m0 x15 h4 y4e ff3 fs2 fc2 sc0 lsb ws56">Seguindo a segunda abordagem e dando um valor arbitrário</div><div class="t m0 x14 h4 y4f ff4 fs2 fc2 sc0 ls2c">t<span class="ff3 lsa ws7">para <span class="_4 blank"> </span></span><span class="lsb">y<span class="ff3 ws3c">, obtemos</span></span></div><div class="t m0 x14 h4 y50 ff3 fs2 fc2 sc0 lsa ws57">Embora estas fórmulas sejam diferentes das obtidas acima,</div><div class="t m0 x14 h4 y51 ff3 fs2 fc2 sc0 lsb ws58">fornecem o mesmo conjunto-solução à medida que <span class="ff4 ls2f">t</span><span class="lsa">varia sobre</span></div><div class="t m0 x14 h4 y52 ff3 fs2 fc2 sc0 lsa ws59">todos os valores reais possíveis. Por exemplo, as fórmulas ante-</div><div class="t m0 x14 h4 y53 ff3 fs2 fc2 sc0 lsa ws5a">riores dão a solução <span class="ff4 lsb ws5b">x = <span class="ff3 ws7">3, <span class="ff4">y </span><span class="ws5c">= quando <span class="_9 blank"></span><span class="ff4 ls30">t<span class="ff3 lsc ws5d">= 3, enquanto as fór-</span></span></span></span></span></div><div class="t m0 x14 h4 y54 ff3 fs2 fc2 sc0 lsb ws3c">mulas acima dão esta solução para <span class="ff4 ls2c">t</span><span class="lsa ws5e">= .</span></div><div class="t m0 x14 h4 y55 ff4 fs2 fc1 sc0 lsb ws7">Solução <span class="ff3">(</span>b<span class="ff3 ls72 ws20">). <span class="fc2 lsa ws5f">Para encontrar o conjunto-solução de (b), nós</span></span></div><div class="t m0 x14 h4 y56 ff3 fs2 fc2 sc0 lsa ws60">podemos atribuir valores arbitrários a quaisquer duas variáveis</div><div class="t m0 x14 h4 y57 ff3 fs2 fc2 sc0 lsa ws61">e resolver na terceira variável. Em particular<span class="_0 blank"></span>, dando os valores</div><div class="t m0 x14 h4 y58 ff3 fs2 fc2 sc0 lsa ws7">arbitrários <span class="_3 blank"> </span><span class="ff4 ls31">s<span class="ff3">e</span>t</span>para <span class="_4 blank"> </span><span class="ff4 lsb ws17">x</span><span class="fs3 ls31 v6">2<span class="fs2 v7">e<span class="ff4 lsb ws17">x</span></span><span class="lsb ws16">3<span class="fs2 ws62 v7">, respectivamente, e resolvendo em</span></span></span></div><div class="t m0 x14 h4 y59 ff4 fs2 fc2 sc0 lsb ws17">x<span class="ff3 fs3 ws16 v6">1</span><span class="ff3 lsa ws6">, nós obtemos</span></div><div class="t m0 x16 h14 y5a ffe fs2 fc1 sc0 lsb">®</div><div class="t m0 x14 he y5b ff1 fs7 fc1 sc0 ls71 ws63">Sistemas Lineares <span class="ff3 fs2 fc2 lsb ws64">Um conjunto finito de equações li-</span></div><div class="t m0 x14 h4 y5c ff3 fs2 fc2 sc0 lsa ws65">neares nas variáveis <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">1</span></span><span class="ls32">,<span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws66">, ..., </span><span class="ls1f">x<span class="fs3 ls33 v6">n</span></span><span class="ff3 ws66">é chamado um </span></span></span><span class="ffd">sistema de</span></div><div class="t m0 x14 h4 y5d ffd fs2 fc2 sc0 lsa ws67">equações lineares <span class="ff3 lsb ws68">ou um </span>sistema linear<span class="ff3 lsb ws68">. Uma seqüência de</span></div><div class="t m0 x14 h4 y5e ff3 fs2 fc2 sc0 lsa ws7">números <span class="_3 blank"> </span><span class="ff4 lsb ws17">s<span class="ff3 fs3 ws16 v6">1</span></span><span class="ls34">,<span class="ff4 ls35">s</span><span class="fs3 lsb ws16 v6">2</span><span class="lsb ws69">, . . ., <span class="ff4 ls36">s<span class="fs3 ls37 v6">n</span></span><span class="ws6a">é chamada uma </span></span></span><span class="ffd ws21">solução </span><span class="ws69">do sistema se </span></div><div class="t m0 x14 h4 y5f ff4 fs2 fc2 sc0 lsb ws17">x<span class="ff3 fs3 ws7 v6">1 <span class="_4 blank"> </span></span><span class="ff3 lsa ws7">= <span class="_3 blank"> </span></span><span class="ls38">s</span><span class="ff3 fs3 ws16 v6">1</span><span class="ff3 ls39">,</span><span class="ls1f">x<span class="ff3 fs3 ls3a v6">2</span><span class="ff3 lsa ws7">= <span class="_4 blank"> </span></span><span class="ls38">s</span></span><span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws6b">, . . ., </span><span class="ls3b">x<span class="fs3 ls3a v6">n</span><span class="ff3 lsa ws7">= <span class="_3 blank"> </span></span><span class="ls38">s<span class="fs3 ls3c v6">n</span></span></span><span class="ff3 ws6b">é uma solução de cada equação do</span></div><div class="t m0 x14 h4 y60 ff3 fs2 fc2 sc0 lsa ws6">sistema. Por exemplo, o sistema</div><div class="t m0 x14 h4 y61 ff3 fs2 fc2 sc0 lsb ws12">tem a solução <span class="ff4 ls3d">x</span><span class="fs3 ws7 v6">1 </span></div><div class="t m0 x17 h4 y62 ff3 fs2 fc2 sc0 lsb ws12">= 1, <span class="ff4 ls3b">x</span><span class="fs3 ls3e v6">2</span>= 2, <span class="ff4 ws17">x</span><span class="fs3 ls3e v6">3</span><span class="lsa ws6c">= \u20131 pois estes valores satisfazem</span></div><div class="t m0 x14 h4 y63 ff3 fs2 fc2 sc0 lsb ws6d">ambas equações. No entanto, <span class="ff4 ws17">x</span><span class="fs3 ws7 v6">1 <span class="_a blank"> </span></span>= 1, <span class="ff4 ls3f">x</span><span class="fs3 ls40 v6">2</span>= 8, <span class="ff4 ws17">x</span><span class="fs3 ls41 v6">3</span>= 1 não é uma</div><div class="t m0 x14 h4 y64 ff3 fs2 fc2 sc0 lsa ws4b">solução do sistema pois estes valores satisfazem apenas a</div><div class="t m0 x14 h4 y65 ff3 fs2 fc2 sc0 lsa ws6">primeira das duas equações do sistema.</div><div class="t m0 x15 h4 y66 ff3 fs2 fc2 sc0 lsa ws6e">Nem todos os sistemas de equações lineares têm solução.</div><div class="t m0 x14 h4 y67 ff3 fs2 fc2 sc0 lsb ws3c">Por exemplo, multiplicando a segunda equação do sistema</div><div class="t m0 x14 h4 y68 ff3 fs2 fc2 sc0 lsb ws6f">por <span class="_b blank"> </span>, torna-se evidente que não existem soluções, pois o sis-</div><div class="t m0 x14 h4 y69 ff3 fs2 fc2 sc0 lsb ws6">tema equivalente</div><div class="t m0 x14 h4 y6a ff3 fs2 fc2 sc0 lsb ws6">que resulta tem equações contraditórias.</div><div class="t m0 x15 h4 y6b ff3 fs2 fc2 sc0 lsb ws70">Um sistema de equações que não possui solução é chamado</div><div class="t m0 x14 h4 y6c ffd fs2 fc2 sc0 lsa">inconsistente<span class="ff3 lsb ws71">; se existir pelo menos uma solução do sistema,</span></div><div class="t m0 x14 h4 y6d ff3 fs2 fc2 sc0 lsb ws72">dizemos que ele é <span class="ffd lsa">consistente<span class="ff3 ws73">. Para ilustrar as possibilidades</span></span></div><div class="t m0 x14 h4 y6e ff3 fs2 fc2 sc0 lsb ws74">que podem ocorrer na resolução de sistemas de equações li-</div><div class="t m0 x14 h4 y6f ff3 fs2 fc2 sc0 lsa ws75">neares, considere um sistema arbitrário de duas equações li-</div><div class="t m0 x14 h4 y70 ff3 fs2 fc2 sc0 lsa ws6">neares nas incógnitas <span class="ff4 ls42">x</span><span class="ls2c">e<span class="ff4 lsb">y<span class="ff3">:</span></span></span></div><div class="t m0 x18 h15 y71 ff3 fsa fc2 sc0 lsb">(<span class="ffd ws22">a</span><span class="ff2 fsb ws23 v8">1</span><span class="ff2 ws7">, </span><span class="ffd ws22">b<span class="ff2 fsb ls43 v8">1</span></span><span class="ff2 ls73 ws76">não ambas nulas</span>)</div><div class="t m0 x18 h15 y72 ff3 fsa fc2 sc0 lsb">(<span class="ffd ws22">a</span><span class="ff2 fsb ws23 v8">2</span><span class="ff2 ws7">, </span><span class="ffd ws22">b<span class="ff2 fsb ls43 v8">2</span></span><span class="ff2 ls73 ws76">não ambas nulas</span>)</div><div class="t m0 x14 h4 y73 ff3 fs2 fc2 sc0 lsa ws77">Os gráficos destas equações são retas, digamos, <span class="ff4 lsb ws17">l</span><span class="fs3 ls74 ws7 v6">1 </span></div><div class="t m0 x19 h4 y73 ff3 fs2 fc2 sc0 lsb ws7">e <span class="_c blank"></span><span class="ff4 ws17">l<span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws78">. Como um</span></span></div><div class="t m0 x14 h4 y74 ff3 fs2 fc2 sc0 lsa ws79">ponto (<span class="ff4 lsb">x</span><span class="ls44">,<span class="ff4 lsb">y</span></span>) está na reta se, e somente se, os números <span class="ff4 ls45">x<span class="ff3">e</span>y</span><span class="lsc">sa-</span></div><div class="t m0 x14 h4 y75 ff3 fs2 fc2 sc0 lsb ws7a">tisfazem a equação da reta, as soluções do sistema de equações</div><div class="t m0 x14 h4 y76 ff3 fs2 fc2 sc0 lsb ws7b">correspondem a pontos de corte de <span class="ff4 ws17">l</span><span class="fs3 ws7 v6">1 </span><span class="ws7">e <span class="_a blank"> </span><span class="ff4 ws17">l</span><span class="fs3 ws16 v6">2</span><span class="lsa ws7c">. Existem três possibi-</span></span></div><div class="t m0 x14 h4 y77 ff3 fs2 fc2 sc0 lsa ws6">lidades, ilustradas na Figura 1.1.1:</div><div class="t m0 x1a h16 y78 fff fs5 fc3 sc0 lsb ws24">a<span class="ff10 fsc ls46 v9">2</span><span class="ls3">x</span><span class="ff11">+</span></div><div class="c x1b y79 w5 h17"><div class="t m0 x1c h16 y7a fff fs5 fc3 sc0 lsb">b</div></div><div class="t m0 x1d h18 y7b ff10 fsc fc3 sc0 ls47">2<span class="fff fs5 ls5 va">y<span class="ff11 ls6">=<span class="fff lsb ws24">c</span></span></span><span class="lsb">2</span></div><div class="t m0 x1a h16 y7c ff12 fs5 fc3 sc0 ls36">a<span class="ff13 fsc ls48 v9">1</span><span class="ls3 v0">x<span class="ff14 ls4">+</span><span class="ls49">b<span class="ff13 fsc ls48 v9">1</span><span class="ls5">y<span class="ff14 ls6">=</span><span class="lsb ws24">c<span class="ff13 fsc v9">1</span></span></span></span></span></div><div class="t m0 x1e ha y7d ff15 fs5 fc3 sc0 ls3">x<span class="ff16 ls4">+</span><span class="ls5">y<span class="ff16 ls6">=<span class="ff17 lsb">4</span></span></span></div><div class="t m0 x1e ha y7e ff15 fs5 fc3 sc0 ls3">x<span class="ff16 ls4">+</span><span class="ls5">y<span class="ff16 ls6">=<span class="ff17 lsb">3</span></span></span></div><div class="t m0 x15 h19 y7f ff18 fsd fc3 sc0 lsb">1</div><div class="t m0 x15 h19 y80 ff18 fsd fc3 sc0 lsb">2</div><div class="t m0 x1e ha y81 ff19 fs5 fc3 sc0 ls5">x<span class="ff1a ls4a">+</span>y<span class="ff1a ls6">=<span class="ff1b lsb">4</span></span></div><div class="t m0 x1f ha y82 ff1b fs5 fc3 sc0 ls1">2<span class="ff19 ls5">x<span class="ff1a ls6">+</span></span>2<span class="ff19 ls5">y<span class="ff1a ls4b">=</span></span><span class="lsb">6</span></div><div class="t m0 x20 ha y83 ff1c fs5 fc3 sc0 ls2">4<span class="ff1d ls4c">x</span><span class="fsc ls4d v9">1</span><span class="ff1e ls4">\u2212<span class="ff1d ls4e">x</span></span><span class="fsc ls4f v9">2</span><span class="ff1e ls4">+</span><span class="lsb">3<span class="ff1d ls50">x</span><span class="fsc ls51 v9">3</span><span class="ff1e ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff1c lsb">1</span></span></span></div><div class="t m0 x20 h1a y84 ff1c fs5 fc3 sc0 lsb">3<span class="ff1d ls52">x</span><span class="fsc ls4d v9">1</span><span class="ff1e ls4 v0">+<span class="ff1d ls4e">x</span></span><span class="fsc ls4f v9">2</span><span class="ff1e ls4 v0">+</span><span class="v0">9<span class="ff1d ls50">x</span><span class="fsc ls51 v9">3</span><span class="ff1e ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff1c lsb">4</span></span></span></div><div class="t m0 x21 ha y85 ff1f fs5 fc3 sc0 ls36">x<span class="ff20 fsc ls53 v9">1</span><span class="ff21 ls6">=<span class="ff20 ls4">5<span class="ff21">+</span><span class="ls2">4</span></span></span><span class="ls3">s<span class="ff21 ls4">\u2212<span class="ff20 lsb">7</span></span><span class="ls54 ws26">t, x</span></span></div><div class="t m0 x22 h1b y86 ff20 fsc fc3 sc0 ls55">2<span class="ff21 fs5 ls6 va">=<span class="ff1f ls56 ws27">s, x</span></span></div><div class="t m0 x23 h1b y86 ff20 fsc fc3 sc0 ls57">3<span class="ff21 fs5 ls6 va">=<span class="ff1f lsb">t</span></span></div><div class="t m0 x24 h19 y87 ff22 fsd fc3 sc0 lsb">11</div><div class="t m0 x25 h19 y88 ff22 fsd fc3 sc0 lsb">2</div><div class="t m0 x26 h19 y89 ff23 fsd fc3 sc0 lsb">11</div><div class="t m0 x27 h19 y8a ff23 fsd fc3 sc0 lsb">2</div><div class="t m0 x20 h1c y8b ff24 fs5 fc3 sc0 ls5">x<span class="ff25 ls58">=<span class="ff26 fsc lsb v5">1</span></span></div><div class="t m0 x28 h1d y8c ff26 fsc fc3 sc0 ls59">2<span class="ff24 fs5 ls5a v5">t<span class="ff25 ls5b">+</span></span><span class="lsb vb">1</span></div><div class="t m0 x26 h1e y8c ff26 fsc fc3 sc0 ls5c">4<span class="ff24 fs5 ls76 ws28 v5">,y<span class="_e blank"></span><span class="ff25 ls6">=<span class="ff24 lsb">t</span></span></span></div><div class="t m0 x20 h1f y8d ff27 fse fc3 sc0 ls5d">\u2212<span class="ff28 fsd lsb vc">3</span></div><div class="t m0 x29 h19 y8e ff28 fsd fc3 sc0 lsb">2</div><div class="c x20 y8f w6 h20"><div class="t m0 xa h21 y90 ff29 fse fc3 sc0 lsb">.</div></div><div class="t m0 x21 h22 y91 ff2a fse fc3 sc0 ls5e">\u2212<span class="ff2b fsd lsb vc">1</span></div><div class="t m0 x17 h19 y92 ff2b fsd fc3 sc0 lsb">2</div><div class="c x21 y8f w7 h23"><div class="t m0 x2a h21 y93 ff2c fse fc3 sc0 lsb">l</div></div><div class="t m0 x2b h24 y94 ff2d fse fc3 sc0 ls5f">\u2212<span class="ff2e fsd lsb vc">1</span></div><div class="t m0 x2c h19 y95 ff2e fsd fc3 sc0 lsb">2</div><div class="c x2d y96 w8 h25"><div class="t m0 x2e h21 y97 ff2f fse fc3 sc0 lsb">y</div></div><div class="t m0 x2d h19 y98 ff30 fsd fc3 sc0 lsb">11</div><div class="t m0 x2f h19 y99 ff30 fsd fc3 sc0 lsb">2</div><div class="t m0 x30 h26 y9a ff31 fs5 fc3 sc0 ls5">x<span class="ff32 ls6">=</span><span class="ls54 ws29">t,<span class="_f blank"> </span>y <span class="ff32 ls6">=<span class="ff33 ls1">2</span></span><span class="ls5a">t<span class="ff32 ls60">\u2212<span class="ff33 fsc lsb v5">1</span></span></span></span></div><div class="t m0 x31 h27 y9b ff33 fsc fc3 sc0 lsb">2</div><div class="t m0 x2 he y9c ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 2<span class="_10 blank"> </span><span class="fs1 fc2 ls77 ws7e">Encontrando um Conjunto-Solução</span></div><div class="t m0 x32 h28 y9d ff34 fs5 fc3 sc0 ls3">x<span class="ff35 ls4">+<span class="ff36 lsb ws2a">3</span><span class="ls61 vd">\u221a</span></span><span class="ls5">y<span class="ff35 ls6">=<span class="ff36 lsb">5</span></span><span class="ls62">,<span class="ff36 lsb">3</span></span></span>x<span class="ff35 ls4">+<span class="ff36 ls1">2</span></span>y<span class="ff35 ls4">\u2212</span><span class="ls63">z<span class="ff35 ls4">+</span><span class="ls78 ws2b">xz <span class="ff35 ls6">=<span class="ff36 ls64">4<span class="ff3 ls62">e</span></span></span><span class="ls5">y<span class="ff35 ls6">=<span class="ff36 lsb">s<span class="ff3">e</span><span class="ls65">n</span><span class="ff34">x</span></span></span></span></span></span></div><div class="t m0 x33 h1c y9e ff37 fs5 fc3 sc0 ls3">x<span class="ff38 ls4">+<span class="ff39 lsb">3</span></span><span class="ls5">y<span class="ff38 ls6">=<span class="ff39 lsb">7</span></span><span class="ls79 ws2c">,y<span class="_11 blank"></span><span class="ff38 ls66">=<span class="ff39 fsc lsb v5">1</span></span></span></span></div><div class="t m0 x34 h29 y9f ff39 fsc fc3 sc0 ls59">2<span class="ff37 fs5 ls3 v5">x<span class="ff38 ls4">+<span class="ff39 lsb">3<span class="ff37 ls63">z</span></span>+<span class="ff39 ls67">1<span class="ff3 ls62">e<span class="ff37 ls68">x</span></span></span></span></span><span class="ls4d ve">1</span><span class="ff38 fs5 ls4 v5">\u2212<span class="ff39 ls1">2<span class="ff37 ls69">x</span></span></span><span class="ls4f ve">2</span><span class="ff38 fs5 ls4 v5">\u2212<span class="ff39 lsb">3<span class="ff37 ls50">x</span></span></span><span class="ls4d ve">3</span><span class="ff38 fs5 ls4 v5">+<span class="ff37 ls4e">x</span></span><span class="ls6a ve">4</span><span class="ff38 fs5 ls6 v5">=<span class="ff39 lsb">7</span></span></div><div class="t m0 x2 he ya0 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 1<span class="_10 blank"> </span><span class="fs1 fc2 ls7a ws7e">Equações Lineares</span></div><div class="t m0 x35 h16 ya1 ff3a fs5 fc3 sc0 lsb ws24">a<span class="ff3b fsc ls48 v9">1</span><span class="ls36">x<span class="ff3b fsc ls6b v9">1</span><span class="ff3c ls4">+</span><span class="ls4e">a<span class="ff3b fsc ls46 v9">2</span></span>x<span class="ff3b fsc ls4f v9">2</span><span class="ff3c ls7b ws2d">+·<span class="_0 blank"></span>·<span class="_0 blank"></span>·+<span class="ff3a lsb ws24">a<span class="fsc ls6c v9">n</span>x<span class="fsc ls6d v9">n</span></span><span class="ls6">=<span class="ff3a lsb">b</span></span></span></span></div><div class="t m0 x36 h16 ya2 ff3d fs5 fc3 sc0 lsb ws24">a<span class="ff3e fsc ls48 v9">1</span><span class="ls3">x<span class="ff3f ls4">+</span><span class="ls6e">a<span class="ff3e fsc ls47 v9">2</span><span class="ls5">y<span class="ff3f ls6">=</span></span></span></span>b</div><div class="t m0 xb h2a ya3 ff1 fs7 fc1 sc0 lsd ws2e">28 <span class="fsf ls7c ws7f vf">\u2022 \u2022 \u2022<span class="_12 blank"> </span></span><span class="ff3 fs10 fc2 ls7d ws80">Álgebra Linear com <span class="_0 blank"></span>Aplicações</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb ya4 w9 h2b" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg3.png"><div class="t m0 x13 h14 ya5 ffe fs2 fc1 sc0 ls7e">\u2211<span class="ff3 fc2 lsa ws4e">As retas <span class="ff4 lsb ws17">l<span class="ff3 fs3 ws7 v6">1 </span><span class="ff3 ws7">e </span><span class="ls7f">l</span></span><span class="fs3 ls80 v6">2</span><span class="lsb ws96">podem ser paralelas, caso em que não há</span></span></div><div class="t m0 x37 h4 ya6 ff3 fs2 fc2 sc0 lsb ws97">interseção e conseqüentemente não existe nenhuma</div><div class="t m0 x37 h4 ya7 ff3 fs2 fc2 sc0 lsa ws6">solução do sistema.</div><div class="t m0 x13 h14 ya8 ffe fs2 fc1 sc0 ls7e">\u2211<span class="ff3 fc2 lsa ws98">As retas <span class="ff4 lsb ws17">l<span class="ff3 fs3 ws7 v6">1 <span class="_3 blank"> </span><span class="fs2 v7">e <span class="_13 blank"> </span></span></span><span class="v0">l</span></span><span class="fs3 ls81 v6">2</span><span class="lsb v0">podem cortar-se em um único ponto,</span></span></div><div class="t m0 x37 h4 ya9 ff3 fs2 fc2 sc0 lsb ws3c">caso em que o sistema tem exatamente uma solução.</div><div class="t m0 x13 h14 yaa ffe fs2 fc1 sc0 ls7e">\u2211<span class="ff3 fc2 lsa ws99">As retas <span class="ff4 lsb ws17">l<span class="ff3 fs3 ws7 v6">1 </span></span></span></div><div class="t m0 x38 h4 yab ff3 fs2 fc2 sc0 lsb ws7">e <span class="ff4 ws17">l</span><span class="fs3 ls82 v6">2</span><span class="ws99">podem coincidir<span class="_0 blank"></span>, caso em que existe uma</span></div><div class="t m0 x37 h4 yac ff3 fs2 fc2 sc0 lsa ws6">infinidade de soluções do sistema.</div><div class="t m0 x39 h2c yad ff3 fs10 fc2 sc0 lsb">(<span class="ff4">a</span><span class="ls7d ws9a">) Nenhuma solução (</span><span class="ff4">b</span><span class="ls7d ws9a">) Uma solução (</span><span class="ff4">c</span><span class="ls7d ws9a">) Infinitas soluções</span></div><div class="t m0 x39 h2d yae ff1 fs2 fc1 sc0 lsb ws9b">Figuras 1.1.1</div><div class="t m0 xb h4 yaf ff3 fs2 fc2 sc0 lsb ws9c">Embora nós aqui tenhamos considerado apenas duas equações</div><div class="t m0 xb h4 yb0 ff3 fs2 fc2 sc0 lsa ws9d">em duas incógnitas, nós mostraremos mais tarde que as mesmas</div><div class="t m0 xb h4 yb1 ff3 fs2 fc2 sc0 lsa ws9e">três possibilidades valem para sistemas de equações lineares</div><div class="t m0 xb h4 yb2 ff3 fs2 fc2 sc0 lsa">arbitrários:</div><div class="t m0 x13 h4 yb3 ff3 fs2 fc2 sc0 lsa ws9f">Um sistema arbitrário de <span class="ff4 ls83">m</span><span class="wsa0">equações lineares em <span class="ff4 ls83">n</span><span class="lsb">incóg-</span></span></div><div class="t m0 xb h4 yb4 ff3 fs2 fc2 sc0 lsa ws6">nitas pode ser escrito como</div><div class="t m0 xb h4 yb5 ff3 fs2 fc2 sc0 lsb ws7">onde <span class="_3 blank"> </span><span class="ff4 ws17">x</span><span class="fs3 ws16 v6">1</span><span class="ls84 v0">,<span class="ff4 lsb ws17">x</span></span><span class="fs3 ws16 v6">2</span><span class="wsa1 v0">, ..., <span class="ff4 ws17">x<span class="fs3 ls85 v6">n</span></span><span class="lsa">são as incógnitas e as letras <span class="ff4 ls86">a</span></span><span class="ws7">e <span class="_3 blank"> </span><span class="ff4 ls84">b</span></span><span class="lsa">com sub-</span></span></div><div class="t m0 xb h4 yb6 ff3 fs2 fc2 sc0 lsa wsa2">scritos denotam constantes. Por exemplo, um sistema geral de</div><div class="t m0 xb h4 yb7 ff3 fs2 fc2 sc0 lsa wsa3">três equações lineares em quatro incógnitas pode ser escrito</div><div class="t m0 xb h4 yb8 ff3 fs2 fc2 sc0 lsb">como</div><div class="t m0 x13 h4 yb9 ff3 fs2 fc2 sc0 lsa wsa4">O subscrito duplo nos coeficientes das incógnitas é um</div><div class="t m0 xb h4 yba ff3 fs2 fc2 sc0 lsb wsa5">recurso útil que é usado para especificar a localização do coefi-</div><div class="t m0 xb h2e ybb ff3 fs2 fc2 sc0 lsa wsa6">ciente no sistema. O primeiro subscrito no coeficiente <span class="ff4 lsb ws17">a<span class="fs3 wsa7 v6">i j<span class="_14 blank"> </span></span><span class="ff3 v0">indica</span></span></div><div class="t m0 xb h4 ybc ff3 fs2 fc2 sc0 lsb wsa8">a equação na qual o coeficiente ocorre e o segundo subscrito</div><div class="t m0 xb h4 ybd ff3 fs2 fc2 sc0 lsb wsa9">indica qual incógnita ele multiplica. <span class="_0 blank"></span>Assim, <span class="ff4 ws17">a</span><span class="fs3 ls74 ws81 v6">12 </span><span class="lsa">ocorre na</span></div><div class="t m0 xb h4 ybe ff3 fs2 fc2 sc0 lsb ws6">primeira equação e multiplica a incógnita <span class="ff4 ws17">x</span><span class="fs3 ws16 v6">2</span>.</div><div class="t m0 xb he ybf ff1 fs7 fc1 sc0 ls71 wsaa">Matrizes Aumentadas <span class="ff3 fs2 fc2 lsa wsab">Se nós mantivermos guardado na</span></div><div class="t m0 xb h4 yc0 ff3 fs2 fc2 sc0 lsb wsac">memória a localização dos sinais de soma, das variáveis e das</div><div class="t m0 xb h4 yc1 ff3 fs2 fc2 sc0 lsa wsad">constantes, poderemos abreviar a escrita de um sistema de <span class="ff4 lsb">m</span></div><div class="t m0 xb h4 yc2 ff3 fs2 fc2 sc0 lsa ws6">equações lineares em <span class="ff4 ls87">n</span><span class="lsb ws3c">incógnitas para:</span></div><div class="t m0 x14 h4 yc3 ff3 fs2 fc2 sc0 ls88 wsae">Esta é chamada a <span class="ffd">matriz aumentada </span>do sistema. (Em</div><div class="t m0 x14 h4 yc4 ff3 fs2 fc2 sc0 lsb6 wsaf">Matemática, o termo <span class="ff4 lsb7 ws82">matriz </span>é utilizado para denotar uma coleção</div><div class="t m0 x14 h4 yc5 ff3 fs2 fc2 sc0 lsb8 wsb0">retangular de números. <span class="_0 blank"></span>As matrizes surgem em vários contextos,</div><div class="t m0 x14 h4 yc6 ff3 fs2 fc2 sc0 lsb8 wsb1">que consideraremos com mais detalhes em seções posteriores.)</div><div class="t m0 x14 h4 yc7 ff3 fs2 fc2 sc0 lsb6 ws3c">Por exemplo, a matriz aumentada do sistema de equações</div><div class="t m0 x14 h4 yc8 ff3 fs2 fc2 sc0 lsb">é</div><div class="t m0 x14 h4 yc9 ff3 fs11 fc1 sc0 ls89 ws83">OBSER<span class="_0 blank"></span>V<span class="_5 blank"></span>AÇÃO<span class="fs2 ls8a">.<span class="fc2 lsb9 wsb2">Quando construímos a matriz aumentada, as</span></span></div><div class="t m0 x14 h4 yca ff3 fs2 fc2 sc0 lsa wsb3">incógnitas devem estar escritas na mesma ordem em cada</div><div class="t m0 x14 h4 ycb ff3 fs2 fc2 sc0 lsb wsb4">equação e as constantes que não multiplicam incógnitas devem</div><div class="t m0 x14 h4 ycc ff3 fs2 fc2 sc0 lsa ws6">estar à direita.</div><div class="t m0 x15 h4 ycd ff3 fs2 fc2 sc0 lsb wsb5">O método básico de resolver um sistema de equações li-</div><div class="t m0 x14 h4 yce ff3 fs2 fc2 sc0 lsba wsb6">neares é substituir o sistema dado por um sistema novo que tem</div><div class="t m0 x14 h4 ycf ff3 fs2 fc2 sc0 lsa wsb7">o mesmo conjunto-solução mas que é mais simples de resolver<span class="_0 blank"></span>.</div><div class="t m0 x14 h4 yd0 ff3 fs2 fc2 sc0 lsa wsb8">Este sistema novo é geralmente obtido numa sucessão de passos</div><div class="t m0 x14 h4 yd1 ff3 fs2 fc2 sc0 lsa wsb9">aplicando os seguintes três tipos de operações para eliminar sis-</div><div class="t m0 x14 h4 yd2 ff3 fs2 fc2 sc0 lsb ws3c">tematicamente as incógnitas.</div><div class="t m0 x15 h4 yd3 ff2 fs2 fc1 sc0 lsb ws84">1. <span class="ff3 fc2 ws3c">Multiplicar uma equação inteira por uma constante</span></div><div class="t m0 x3a h4 yd4 ff3 fs2 fc2 sc0 lsb">não-nula.</div><div class="t m0 x15 h4 yd5 ff2 fs2 fc1 sc0 lsb ws84">2. <span class="ff3 fc2 lsa ws6">T<span class="_0 blank"></span>rocar duas equações entre si.</span></div><div class="t m0 x15 h4 yd6 ff2 fs2 fc1 sc0 lsb ws84">3. <span class="ff3 fc2 ws3c">Somar um múltiplo de uma equação a uma outra</span></div><div class="t m0 x3a h4 yd7 ff3 fs2 fc2 sc0 lsb">equação.</div><div class="t m0 x14 h4 yd8 ff3 fs2 fc2 sc0 lsb wsba">Como as linhas (horizontais) de uma matriz aumentada cor-</div><div class="t m0 x14 h4 yd9 ff3 fs2 fc2 sc0 lsa ws98">respondem às equações no sistema associado, estas três ope-</div><div class="t m0 x14 h4 yda ff3 fs2 fc2 sc0 lsa wsbb">rações correspondem às seguintes operações nas linhas da</div><div class="t m0 x14 h4 ydb ff3 fs2 fc2 sc0 lsb ws3c">matriz aumentada.</div><div class="t m0 x15 h4 ydc ff2 fs2 fc1 sc0 lsb ws84">1. <span class="ff3 fc2 ws3c">Multiplicar uma linha inteira por uma constante não-</span></div><div class="t m0 x3a h4 ydd ff3 fs2 fc2 sc0 lsb">nula.</div><div class="t m0 x15 h4 yde ff2 fs2 fc1 sc0 lsb ws84">2. <span class="ff3 fc2 lsa ws6">T<span class="_0 blank"></span>rocar duas linhas entre si.</span></div><div class="t m0 x15 h4 ydf ff2 fs2 fc1 sc0 lsb ws84">3. <span class="ff3 fc2 ws3c">Somar um múltiplo de uma linha a uma outra linha.</span></div><div class="t m0 x14 he ye0 ff1 fs7 fc1 sc0 lsbb wsbc">Operações Elementares sobre Linhas <span class="ff3 fs2 fc2 lsa wsbd">Estas três</span></div><div class="t m0 x14 h4 ye1 ff3 fs2 fc2 sc0 lsa wsbe">operações são chamadas <span class="ffd">operações elementares sobre linhas</span><span class="lsb">.</span></div><div class="t m0 x14 h4 ye2 ff3 fs2 fc2 sc0 lsa wsbf">O seguinte exemplo ilustra como estas operações podem ser</div><div class="t m0 x14 h4 ye3 ff3 fs2 fc2 sc0 lsa wsc0">usadas para resolver sistemas de equações lineares. Como na</div><div class="t m0 x14 h4 ye4 ff3 fs2 fc2 sc0 lsa wsc1">próxima seção iremos desenvolver um procedimento sistemáti-</div><div class="t m0 x14 h4 ye5 ff3 fs2 fc2 sc0 lsa wsc2">co para encontrar soluções, não é preciso ficar preocupado sobre</div><div class="t m0 x14 h4 ye6 ff3 fs2 fc2 sc0 lsa wsc3">o porquê dos passos tomados neste exemplo. O esforço aqui</div><div class="t m0 x14 h4 ye7 ff3 fs2 fc2 sc0 lsa ws6">deveria ser para entender as contas e a discussão.</div><div class="t m0 x14 h4 ye8 ff3 fs2 fc2 sc0 lsa wsc4">Na coluna da esquerda nós resolvemos um sistema de equações</div><div class="t m0 x14 h4 ye9 ff3 fs2 fc2 sc0 lsb wsc5">lineares operando nas equações do sistema e na coluna da di-</div><div class="t m0 x14 h4 yea ff3 fs2 fc2 sc0 lsa wsc6">reita nós resolvemos o mesmo sistema operando nas linhas da</div><div class="t m0 x14 h4 yeb ff3 fs2 fc2 sc0 lsb ws3c">matriz aumentada.</div><div class="t m0 x3b ha yec ff40 fs5 fc3 sc0 ls5">x<span class="ff41 ls8b">+</span>y<span class="ff41 ls6">+<span class="ff42 ls1">2</span></span><span class="ls8c">z<span class="ff41 ls6">=<span class="ff42 lsb">9</span></span></span></div><div class="t m0 x3c ha yed ff42 fs5 fc3 sc0 ls1">2<span class="ff40 ls5">x<span class="ff41 ls6">+</span></span><span class="ls2">4<span class="ff40 ls5">y<span class="ff41 ls6">\u2212</span></span><span class="lsb">3<span class="ff40 ls8d">z<span class="ff41 ls6">=</span></span>1</span></span></div><div class="t m0 x3c ha yee ff42 fs5 fc3 sc0 lsb">3<span class="ff40 ls5">x<span class="ff41 ls4b">+</span></span>6<span class="ff40 ls5">y<span class="ff41 ls6">\u2212</span></span>5<span class="ff40 ls8d">z<span class="ff41 ls6">=</span></span>0</div><div class="t m0 x24 h9 yef ff43 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x24 h9 yf0 ff43 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x24 h9 yf1 ff43 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x3d ha yec ff42 fs5 fc3 sc0 lsbc">1129</div><div class="t m0 x3d ha yed ff42 fs5 fc3 sc0 lsbd ws85">24<span class="_15 blank"></span><span class="ff41 lsb">\u2212<span class="ff42 lsbe">31</span></span></div><div class="t m0 x3d ha yee ff42 fs5 fc3 sc0 lsbe ws86">36<span class="_15 blank"></span><span class="ff41 lsb">\u2212<span class="ff42 lsbf">50</span></span></div><div class="t m0 x3e h9 yef ff43 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x3e h9 yf0 ff43 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x3e h9 yf1 ff43 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x3f he yf2 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 3<span class="_10 blank"> </span><span class="fs1 fc2 ls9 ws7e">Usando Operações Elementares</span></div><div class="t m0 x40 h3 yf3 ff1 fs1 fc2 sc0 ls7a ws7e">sobre Linhas</div><div class="t m0 x20 h9 yf4 ff44 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x20 h9 yf5 ff44 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x41 ha yf6 ff45 fs5 fc3 sc0 lsbc">1129</div><div class="t m0 x41 ha yf7 ff45 fs5 fc3 sc0 lsbd ws85">24<span class="_15 blank"></span><span class="ff46 lsb">\u2212<span class="ff45 lsbe">31</span></span></div><div class="t m0 x41 ha yf8 ff45 fs5 fc3 sc0 lsbe ws86">36<span class="_15 blank"></span><span class="ff46 lsb">\u2212<span class="ff45 lsbf">50</span></span></div><div class="t m0 x24 h9 yf4 ff44 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x24 h9 yf5 ff44 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x42 ha yf9 ff47 fs5 fc3 sc0 lsb ws24">x<span class="ff48 fsc ls51 v9">1</span><span class="ff49 ls8b">+</span>x<span class="ff48 fsc ls55 v9">2</span><span class="ff49 ls6">+<span class="ff48 ls1">2</span></span>x<span class="ff48 fsc ls51 v9">3</span><span class="ff49 ls6">=</span><span class="ff48">9</span></div><div class="t m0 x20 h2f yfa ff48 fs5 fc3 sc0 ls1">2<span class="ff47 lsb ws24">x</span><span class="fsc ls51 v9">1</span><span class="ff49 ls6 v0">+<span class="ff48 ls2">4<span class="ff47 lsb ws24">x</span><span class="fsc ls55 v9">2</span></span><span class="ls8e">\u2212<span class="ff48 lsb">3<span class="ff47 ws24">x</span><span class="fsc ls51 v9">3</span></span></span>=<span class="ff48 lsb">1</span></span></div><div class="t m0 x20 h1a yfb ff48 fs5 fc3 sc0 lsb">3<span class="ff47 ws24">x</span><span class="fsc ls51 v9">1</span><span class="ff49 ls4b v0">+</span><span class="v0">6<span class="ff47 ws24">x</span><span class="fsc ls8f v9">2</span><span class="ff49 ls8e">\u2212</span>5<span class="ff47 ws24">x</span><span class="fsc ls51 v9">3</span><span class="ff49 ls6">=</span>0</span></div><div class="t m0 x43 h9 yfc ff4a fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x43 h9 yfd ff4a fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x43 h9 yfe ff4a fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x43 h9 yff ff4a fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x43 h9 y100 ff4a fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x44 h30 y101 ff4b fs5 fc3 sc0 ls90">a<span class="ff4c fsc lsb ws87 v9">11 </span><span class="lsb ws24 v0">a<span class="ff4c fsc ws88 v9">12 </span><span class="ff4d ls91 ws89">··· </span><span class="ls92">a</span><span class="ff4c fsc v9">1<span class="ff4b ls93">n</span></span><span class="ls36">b</span><span class="ff4c fsc v9">1</span></span></div><div class="t m0 x44 h31 y102 ff4b fs5 fc3 sc0 ls36">a<span class="ff4c fsc lsb ws87 v9">21 </span><span class="lsb ws24">a<span class="ff4c fsc ws88 v9">22 </span></span><span class="ff4d ls91 ws8a">··· </span><span class="lsb ws24">a<span class="ff4c fsc ls94 v9">2<span class="ff4b ls95">n</span></span><span class="v0">b<span class="ff4c fsc v9">2</span></span></span></div><div class="t m0 x45 h16 y103 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x45 h16 y104 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x45 h32 y105 ff4b fs5 fc3 sc0 ls96">.<span class="lsb v10">.</span></div><div class="t m0 x46 h16 y104 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x46 h32 y105 ff4b fs5 fc3 sc0 ls97">.<span class="lsb v10">.</span></div><div class="t m0 x47 h16 y104 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x47 h32 y105 ff4b fs5 fc3 sc0 ls98">.<span class="lsb v10">.</span></div><div class="t m0 x48 h16 y104 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x48 h16 y105 ff4b fs5 fc3 sc0 lsb">.</div><div class="t m0 x44 h33 y106 ff4b fs5 fc3 sc0 ls99">a<span class="fsc lsb v9">m<span class="ff4c ls9a">1</span></span><span class="lsb ws24 v0">a<span class="fsc v9">m<span class="ff4c ls9b">2</span></span><span class="ff4d ls91 ws8b">··· </span><span class="ls9c">a</span><span class="fsc ws8c v9">mn </span>b<span class="fsc v9">m</span></span></div><div class="t m0 x49 h9 yfc ff4a fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x49 h9 yfd ff4a fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x49 h9 yfe ff4a fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x49 h9 yff ff4a fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x49 h9 y100 ff4a fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x4a h16 y107 ff4e fs5 fc3 sc0 ls3d">a<span class="ff4f fsc lsb ws8d v9">11 </span><span class="lsb ws24">x<span class="ff4f fsc ls4d v9">1</span><span class="ff50 ls4">+</span><span class="ls4e">a</span><span class="ff4f fsc ws8e v9">12 </span><span class="ls36">x<span class="ff4f fsc ls4f v9">2</span><span class="ff50 ls4">+</span><span class="ls4e">a</span></span><span class="ff4f fsc ws8d v9">13 </span>x<span class="ff4f fsc ls4d v9">3</span><span class="ff50 ls4">+</span><span class="ls4e">a</span><span class="ff4f fsc ws8f v9">14 </span>x<span class="ff4f fsc ls9d v9">4</span><span class="ff50 ls6">=</span>b<span class="ff4f fsc v9">1</span></span></div><div class="t m0 x4a h16 y108 ff4e fs5 fc3 sc0 ls36">a<span class="ff4f fsc lsb ws8d v9">21 </span><span class="lsb ws24">x<span class="ff4f fsc ls4d v9">1</span><span class="ff50 ls4">+</span><span class="ls4e">a</span><span class="ff4f fsc ws8e v9">22 </span></span>x<span class="ff4f fsc ls4f v9">2</span><span class="ff50 ls4">+</span><span class="ls4e">a<span class="ff4f fsc lsb ws8d v9">23 </span><span class="lsb ws24">x<span class="ff4f fsc ls4d v9">3</span><span class="ff50 ls4">+</span></span>a<span class="ff4f fsc lsb ws8f v9">24 </span><span class="lsb ws24">x<span class="ff4f fsc ls9d v9">4</span><span class="ff50 ls6">=</span>b<span class="ff4f fsc v9">2</span></span></span></div><div class="t m0 x4a h34 y109 ff4e fs5 fc3 sc0 ls36">a<span class="ff4f fsc lsb ws8d v9">31 </span><span class="lsb ws24">x<span class="ff4f fsc ls4d v9">1</span><span class="ff50 ls4 v0">+<span class="ff4e ls4e">a</span></span><span class="ff4f fsc ws8e v9">32 </span></span><span class="v0">x<span class="ff4f fsc ls4f v9">2</span><span class="ff50 ls4">+</span><span class="ls4e">a<span class="ff4f fsc lsb ws8d v9">33 </span><span class="lsb ws24">x<span class="ff4f fsc ls4d v9">3</span><span class="ff50 ls4">+</span></span>a<span class="ff4f fsc lsb ws8f v9">34 </span><span class="lsb ws24">x<span class="ff4f fsc ls9d v9">4</span></span></span></span><span class="ff50 ls6">=</span><span class="lsb ws24">b<span class="ff4f fsc v9">3</span></span></div><div class="t m0 x4a h16 y10a ff51 fs5 fc3 sc0 ls36">a<span class="ff52 fsc lsb ws8d v9">11 </span><span class="lsb ws24 v0">x<span class="ff52 fsc ls9e v9">1</span><span class="ff53 ls9f">+</span><span class="lsa0">a</span><span class="ff52 fsc ws90 v9">12 </span>x<span class="ff52 fsc lsa1 v9">2</span><span class="ff53 ls91 ws91">+···+ </span>a<span class="ff52 fsc v9">1<span class="ff51 ls6c">n</span></span>x<span class="fsc lsa2 v9">n</span><span class="ff53 lsa3">=</span><span class="lsa4">b</span><span class="ff52 fsc v9">1</span></span></div><div class="t m0 x4a h16 y10b ff51 fs5 fc3 sc0 ls36">a<span class="ff52 fsc lsb ws8d v9">21 </span><span class="lsb ws24">x<span class="ff52 fsc ls9e v9">1</span><span class="ff53 ls9f">+</span><span class="lsa0">a</span><span class="ff52 fsc ws90 v9">22 </span>x<span class="ff52 fsc lsa1 v9">2</span></span><span class="ff53 ls91 ws92">+···+ </span><span class="lsb ws24">a<span class="ff52 fsc ls94 v9">2<span class="ff51 lsa5">n</span></span></span>x<span class="fsc lsa6 v9">n</span><span class="ff53 lsa7">=</span><span class="lsa8">b<span class="ff52 fsc lsb v9">2</span></span></div><div class="t m0 x38 h16 y10c ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x38 h16 y10d ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x38 h32 y10e ff51 fs5 fc3 sc0 lsa9">.<span class="lsb v10">.</span></div><div class="t m0 x4b h16 y10d ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x4b h32 y10e ff51 fs5 fc3 sc0 lsaa">.<span class="lsb v10">.</span></div><div class="t m0 x48 h16 y10d ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x48 h32 y10e ff51 fs5 fc3 sc0 lsab">.<span class="lsb v10">.</span></div><div class="t m0 x4c h16 y10d ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x4c h16 y10e ff51 fs5 fc3 sc0 lsb">.</div><div class="t m0 x4a h33 y10f ff51 fs5 fc3 sc0 lsb ws24">a<span class="fsc v9">m<span class="ff52 lsac">1</span></span><span class="v0">x<span class="ff52 fsc ls53 v9">1</span><span class="ff53 lsad">+</span>a<span class="fsc v9">m<span class="ff52 lsae">2</span></span>x<span class="ff52 fsc ls8f v9">2</span><span class="ff53 ls91 ws93">+···+ </span><span class="lsaf">a</span><span class="fsc ws94 v9">mn </span>x<span class="fsc ls6d v9">n</span><span class="ff53 ls6">=</span>b<span class="fsc v9">m</span></span></div><div class="t m0 x4d h35 y110 ff4 fs2 fc2 sc0 lsb0 wsc7">T<span class="_5 blank"></span>odo sistema de equações linear<span class="_0 blank"></span>es tem ou nenhuma</div><div class="t m0 x4d h35 y111 ff4 fs2 fc2 sc0 lsb wsc8">solução, ou exatamente uma, ou então uma infinidade de</div><div class="t m0 x4d h35 y112 ff4 fs2 fc2 sc0 lsa">soluções.</div><div class="t m0 x4e h36 y113 ff54 fs12 fc3 sc0 lsb">x</div><div class="t m0 x4f h37 y114 ff54 fs12 fc3 sc0 lsb1">y<span class="lsb ws95 v11">l</span><span class="ff55 fs13 lsb2 v12">1</span><span class="ff56 lsb2 v11">e</span><span class="lsb ws95 v11">l<span class="ff55 fs13 v13">2</span></span></div><div class="t m0 x48 h36 y115 ff57 fs12 fc3 sc0 lsb">x</div><div class="t m0 x50 h37 y116 ff57 fs12 fc3 sc0 lsb3">y<span class="lsb ws95 v11">l</span><span class="ff58 fs13 lsb4 v12">1</span><span class="lsb ws95 v11">l<span class="ff58 fs13 v13">2</span></span></div><div class="t m0 x35 h36 y117 ff59 fs12 fc3 sc0 lsb">x</div><div class="t m0 x51 h36 y118 ff59 fs12 fc3 sc0 lsb">y</div><div class="t m0 x52 h36 y119 ff59 fs12 fc3 sc0 lsb ws95">l<span class="ff5a fs13 lsb5 v13">1</span>l<span class="ff5a fs13 v13">2</span></div><div class="t m0 x3f h38 ya3 ff4 fs10 fc2 sc0 ls7d ws9a">Capítulo 1 - Sistemas de Equações Linear<span class="_0 blank"></span>es e Matrizes<span class="_12 blank"> </span><span class="ff1 fsf fc1 ls7c ws7f v14">\u2022 \u2022 \u2022<span class="_12 blank"> </span><span class="fs7 lsd v5">29</span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y25 w9 hf" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg4.png"><div class="t m0 xb h15 y11a ff3 fsa fc2 sc0 lsb ws76">Some \u20132 vezes a primeira equação<span class="_16 blank"> </span>Some \u20132 vezes a primeira linha</div><div class="t m0 xb h15 y11b ff3 fsa fc2 sc0 ls73 ws76">à segunda para obter esquerda<span class="_17 blank"> </span>à segunda para obter direita</div><div class="t m0 xb h15 y11c ff3 fsa fc2 sc0 lsb ws76">Some \u20133 vezes a primeira equação<span class="_16 blank"> </span>Some \u20133 vezes a primeira linha</div><div class="t m0 xb h15 y11d ff3 fsa fc2 sc0 ls73 ws76">à terceira para obter<span class="_18 blank"> </span>à terceira para obter</div><div class="t m0 xb h15 y11e ff3 fsa fc2 sc0 lsb ws76">Multiplique a segunda equação<span class="_19 blank"> </span>Multiplique a segunda linha</div><div class="t m0 xb h15 y11f ff3 fsa fc2 sc0 ls73 wse9">por para <span class="_1a blank"></span>obter<span class="_1b blank"> </span>por para <span class="_1a blank"></span>obter</div><div class="t m0 xb h15 y120 ff3 fsa fc2 sc0 lsb ws76">Some \u20133 vezes a segunda equação<span class="_1c blank"> </span>Some \u20133 vezes a segunda linha</div><div class="t m0 xb h15 y121 ff3 fsa fc2 sc0 ls73 ws76">à terceira para obter<span class="_18 blank"> </span>à terceira para obter</div><div class="t m0 x14 h15 y122 ff3 fsa fc2 sc0 lsb ws76">Multiplique a terceira equação<span class="_1d blank"> </span>Multiplique a terceira linha</div><div class="t m0 x14 h15 y123 ff3 fsa fc2 sc0 ls73 ws76">por \u20132 para obter<span class="_1b blank"> </span>por \u20132 para obter</div><div class="t m0 x14 h15 y124 ff3 fsa fc2 sc0 lsb ws76">Some \u20131 vez a segunda equação<span class="_1e blank"> </span>Some \u20131 vez a segunda linha</div><div class="t m0 x14 h15 y125 ff3 fsa fc2 sc0 ls73 ws76">à primeira para obter<span class="_1f blank"> </span>à primeira para obter</div><div class="t m0 x14 h15 y126 ff3 fsa fc2 sc0 ls73 ws76">Some \u2013 <span class="_20 blank"> </span>vezes a terceira<span class="_21 blank"> </span>Some \u2013 <span class="_20 blank"> </span>vezes a terceira linha</div><div class="t m0 x14 h15 y127 ff3 fsa fc2 sc0 ls73 ws76">equação à primeira e <span class="_22 blank"> </span>vezes<span class="_23 blank"> </span>à primeira e <span class="_22 blank"> </span>vezes a terceira</div><div class="t m0 x14 h15 y128 ff3 fsa fc2 sc0 lsb ws76">a terceira equação à segunda para obter<span class="_24 blank"> </span>equação à segunda para obter</div><div class="t m0 x14 h14 y129 ff3 fs2 fc2 sc0 lsa ws7">A<span class="_25 blank"> </span>solução <span class="ff4 ls2c">x</span><span class="lsb ws6">= 1, <span class="ff4 ls2c">y</span>= 2, <span class="ff4 ls2c">z</span><span class="ws3c">= 3 pode, agora, ser visualizada.<span class="_26 blank"> </span><span class="ffe fc1">®</span></span></span></div><div class="t m0 x53 ha y12a ff5b fs5 fc3 sc0 lsc0">x<span class="ff5c ls8e">=<span class="ff5d lsb">1</span></span></div><div class="t m0 xc ha y12b ff5b fs5 fc3 sc0 lsc1">y<span class="ff5c ls6">=<span class="ff5d lsb">2</span></span></div><div class="t m0 x54 ha y12c ff5b fs5 fc3 sc0 ls8c">z<span class="ff5c ls8e">=<span class="ff5d lsb">3</span></span></div><div class="t m0 x25 h9 y12d ff5e fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x25 h9 y12e ff5e fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x25 h9 y12f ff5e fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x3d ha y130 ff5d fs5 fc3 sc0 lsc2 wsc9">1001</div><div class="t m0 x3d ha y131 ff5d fs5 fc3 sc0 ls130 wsca">0102</div><div class="t m0 x3d ha y132 ff5d fs5 fc3 sc0 ls131 wscb">0013</div><div class="t m0 x3e h9 y12d ff5e fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x3e h9 y12e ff5e fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x3e h9 y12f ff5e fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x55 h39 y133 ff5f fs14 fc3 sc0 lsb">7</div><div class="t m0 x55 h39 y134 ff5f fs14 fc3 sc0 lsb">2</div><div class="t m0 x56 h39 y133 ff60 fs14 fc3 sc0 lsb">7</div><div class="t m0 x56 h39 y134 ff60 fs14 fc3 sc0 lsb">2</div><div class="c x57 y135 wa h3a"><div class="t m0 x2e h3b y136 ff61 fs15 fc3 sc0 lsb">\u2212</div></div><div class="t m0 x57 h39 y137 ff62 fs14 fc3 sc0 lsb">11</div><div class="t m0 x58 h39 y138 ff62 fs14 fc3 sc0 lsb">2</div><div class="c x59 y135 wb h3a"><div class="t m0 x2e h3b y136 ff63 fs15 fc3 sc0 lsb">\u2212</div></div><div class="t m0 x59 h39 y137 ff64 fs14 fc3 sc0 lsb">11</div><div class="t m0 x5a h39 y138 ff64 fs14 fc3 sc0 lsb">2</div><div class="t m0 x3b h1c y139 ff65 fs5 fc3 sc0 lsc3">x<span class="ff66 lsc4">+<span class="ff67 fsc lsb v5">11</span></span></div><div class="t m0 x54 h3c y13a ff67 fsc fc3 sc0 lsc5">2<span class="ff65 fs5 ls8c v5">z<span class="ff66 lsc6">=</span></span><span class="lsb vb">35</span></div><div class="t m0 x5b h27 y13a ff67 fsc fc3 sc0 lsb">2</div><div class="t m0 x5c h3d y13b ff65 fs5 fc3 sc0 ls5">y<span class="ff66 lsc7">\u2212<span class="ff67 fsc lsb v5">7</span></span></div><div class="t m0 x54 h3e y13c ff67 fsc fc3 sc0 ls59">2<span class="ff65 fs5 ls8c v5">z<span class="ff66 ls75">=\u2212</span></span></div><div class="t m0 x5d h27 y13d ff67 fsc fc3 sc0 lsb">17</div><div class="t m0 x5b h27 y13c ff67 fsc fc3 sc0 lsb">2</div><div class="t m0 x1b ha y13e ff65 fs5 fc3 sc0 ls8c">z<span class="ff66 lsc8">=<span class="ff67 lsb">3</span></span></div><div class="t m0 x5e h9 y13f ff68 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x5e h9 y140 ff68 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x5e h9 y141 ff68 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x5e h9 y142 ff68 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x5f ha y143 ff67 fs5 fc3 sc0 lsbe">10</div><div class="t m0 x60 h27 y144 ff67 fsc fc3 sc0 lsb">11</div><div class="t m0 x61 h27 y145 ff67 fsc fc3 sc0 lsb">2</div><div class="t m0 x3e h27 y146 ff67 fsc fc3 sc0 lsb">35</div><div class="t m0 x62 h27 y145 ff67 fsc fc3 sc0 lsb">2</div><div class="t m0 x5f h1c y147 ff67 fs5 fc3 sc0 ls130 wsca">01<span class="_15 blank"></span><span class="ff66 lsc9">\u2212<span class="ff67 fsc lsb v5">7</span></span></div><div class="t m0 x63 h1d y148 ff67 fsc fc3 sc0 lsca">2<span class="ff66 fs5 lscb v5">\u2212</span><span class="lsb vb">17</span></div><div class="t m0 x62 h27 y148 ff67 fsc fc3 sc0 lsb">2</div><div class="t m0 x5f ha y149 ff67 fs5 fc3 sc0 lsc2 wscc">00 1<span class="_2 blank"> </span>3</div><div class="t m0 x64 h9 y14a ff68 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x64 h9 y14b ff68 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x64 h9 y14c ff68 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x64 h9 y14d ff68 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x3b ha y14e ff69 fs5 fc3 sc0 ls5">x<span class="ff6a ls6">+</span>y<span class="ff6a lscc">+<span class="ff6b ls1">2</span></span><span class="ls8c">z<span class="ff6a lsc8">=<span class="ff6b lsb">9</span></span></span></div><div class="t m0 x65 h3f y14f ff69 fs5 fc3 sc0 ls5">y<span class="ff6a lscd">\u2212<span class="ff6b fsc lsb v5">7</span></span></div><div class="t m0 x54 h1e y150 ff6b fsc fc3 sc0 lsce">2<span class="ff69 fs5 ls8c v5">z<span class="ff6a ls75">=\u2212</span></span></div><div class="t m0 x5d h27 y151 ff6b fsc fc3 sc0 lsb">17</div><div class="t m0 x5d h27 y150 ff6b fsc fc3 sc0 lsb">2</div><div class="t m0 x66 ha y152 ff69 fs5 fc3 sc0 ls8c">z<span class="ff6a lsc8">=<span class="ff6b lsb">3</span></span></div><div class="t m0 x5e h9 y153 ff6c fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x5e h9 y154 ff6c fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x5e h9 y155 ff6c fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x5f ha y156 ff6b fs5 fc3 sc0 lsbe wscc">11 2<span class="_27 blank"> </span>9</div><div class="t m0 x5f h40 y157 ff6b fs5 fc3 sc0 lsc2 wsc9">01<span class="_15 blank"></span><span class="ff6a lscf">\u2212<span class="ff6b fsc lsb v5">7</span></span></div><div class="t m0 x61 h3c y158 ff6b fsc fc3 sc0 lsca">2<span class="ff6a fs5 lscb v5">\u2212</span><span class="lsb vb">17</span></div><div class="t m0 x62 h27 y158 ff6b fsc fc3 sc0 lsb">2</div><div class="t m0 x5f ha y159 ff6b fs5 fc3 sc0 lsbe wscd">00 1<span class="_2 blank"> </span>3</div><div class="t m0 x2b h9 y15a ff6c fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x2b h9 y15b ff6c fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x2b h9 y15c ff6c fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x33 ha y15d ff6d fs5 fc3 sc0 ls5">x<span class="ff6e ls6">+</span>y<span class="ff6e ls6">+<span class="ff6f ls1">2</span></span><span class="lsa7">z<span class="ff6e lsc8">=<span class="ff6f lsb">9</span></span></span></div><div class="t m0 x67 h3f y15e ff6d fs5 fc3 sc0 ls5">y<span class="ff6e lscd">\u2212<span class="ff6f fsc lsb v5">7</span></span></div><div class="t m0 x68 h41 y15f ff6f fsc fc3 sc0 lsce">2<span class="ff6d fs5 ls8c v5">z<span class="ff6e ls75">=\u2212</span></span></div><div class="t m0 x69 h27 y160 ff6f fsc fc3 sc0 lsb">17</div><div class="t m0 x6a h27 y15f ff6f fsc fc3 sc0 lsb">2</div><div class="t m0 x11 h1c y161 ff6e fs5 fc3 sc0 lsd0">\u2212<span class="ff6f fsc lsb v5">1</span></div><div class="t m0 x68 h41 y162 ff6f fsc fc3 sc0 lsce">2<span class="ff6d fs5 ls8c v5">z<span class="ff6e ls132">=\u2212</span></span></div><div class="t m0 x6b h27 y163 ff6f fsc fc3 sc0 lsb">3</div><div class="t m0 x6b h27 y162 ff6f fsc fc3 sc0 lsb">2</div><div class="t m0 x49 h9 y164 ff70 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x49 h9 y165 ff70 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x49 h9 y166 ff70 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x49 h9 y167 ff70 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x6c ha y168 ff6f fs5 fc3 sc0 lsbe wscc">11 2<span class="_27 blank"> </span>9</div><div class="t m0 x6c h42 y169 ff6f fs5 fc3 sc0 lsc2 wsc9">01<span class="_15 blank"></span><span class="ff6e lsd1">\u2212<span class="ff6f fsc lsb v5">7</span></span></div><div class="t m0 x6d h1d y16a ff6f fsc fc3 sc0 lsca">2<span class="ff6e fs5 lscb v5">\u2212</span><span class="lsb vb">17</span></div><div class="t m0 x6e h27 y16a ff6f fsc fc3 sc0 lsb">2</div><div class="t m0 x6c h3d y16b ff6f fs5 fc3 sc0 lsbe wsce">00<span class="_15 blank"></span><span class="ff6e lsd2">\u2212<span class="ff6f fsc lsb v5">1</span></span></div><div class="t m0 x6d h1d y16c ff6f fsc fc3 sc0 lsd3">2<span class="ff6e fs5 lsd4 v5">\u2212</span><span class="lsb vb">3</span></div><div class="t m0 x6f h27 y16c ff6f fsc fc3 sc0 lsb">2</div><div class="t m0 x70 h9 y164 ff70 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x70 h9 y165 ff70 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x70 h9 y166 ff70 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x70 h9 y167 ff70 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x52 ha y16d ff71 fs5 fc3 sc0 ls5">x<span class="ff72 lsd5">+</span>y<span class="ff72 lsd6">+<span class="ff73 ls1">2</span></span><span class="ls8c">z<span class="ff72 lsd7">=<span class="ff73 lsb">9</span></span></span></div><div class="t m0 x67 h3f y16e ff71 fs5 fc3 sc0 ls5">y<span class="ff72 lsd8">\u2212<span class="ff73 fsc lsb v5">7</span></span></div><div class="t m0 x71 h3e y16f ff73 fsc fc3 sc0 lsce">2<span class="ff71 fs5 ls8c v5">z<span class="ff72 ls133">=\u2212</span></span></div><div class="t m0 x72 h27 y170 ff73 fsc fc3 sc0 lsb">17</div><div class="t m0 x73 h27 y16f ff73 fsc fc3 sc0 lsb">2</div><div class="t m0 x37 ha y171 ff73 fs5 fc3 sc0 lsb">3<span class="ff71 ls5">y<span class="ff72 ls6">\u2212</span></span>11<span class="ff71 ls8c">z<span class="ff72 ls75 wscf">=\u2212<span class="_d blank"></span><span class="ff73 lsb">27</span></span></span></div><div class="t m0 x74 h9 y172 ff74 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x74 h9 y173 ff74 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x74 h9 y174 ff74 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x75 ha y175 ff73 fs5 fc3 sc0 ls134 wsd0">11 2 9</div><div class="t m0 x75 h43 y176 ff73 fs5 fc3 sc0 lsc2 wsc9">01<span class="_28 blank"></span><span class="ff72 lsd9">\u2212<span class="ff73 fsc lsb v5">7</span></span></div><div class="t m0 x76 h44 y16f ff73 fsc fc3 sc0 lsda">2<span class="ff72 fs5 lscb v5">\u2212</span><span class="lsb vb">17</span></div><div class="t m0 x6f h27 y16f ff73 fsc fc3 sc0 lsb">2</div><div class="t m0 x75 ha y171 ff73 fs5 fc3 sc0 lsc2 wsc9">03<span class="_15 blank"></span><span class="ff72 lsb">\u2212<span class="ff73 wsd1">11 </span>\u2212<span class="ff73">27</span></span></div><div class="t m0 x77 h9 y172 ff74 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x77 h9 y173 ff74 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x77 h9 y174 ff74 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x78 h39 y177 ff75 fs14 fc3 sc0 lsb">1</div><div class="t m0 x78 h39 y178 ff75 fs14 fc3 sc0 lsb">2</div><div class="t m0 x79 h39 y177 ff76 fs14 fc3 sc0 lsb">1</div><div class="t m0 x79 h39 y178 ff76 fs14 fc3 sc0 lsb">2</div><div class="t m0 x52 ha y179 ff77 fs5 fc3 sc0 ls5">x<span class="ff78 ls4a">+</span>y<span class="ff78 lsd6">+<span class="ff79 ls1">2</span></span><span class="ls8c">z<span class="ff78 lsd7">=<span class="ff79 lsb">9</span></span></span></div><div class="t m0 x37 ha y17a ff79 fs5 fc3 sc0 ls1">2<span class="ff77 ls5">y<span class="ff78 lsd5">\u2212</span></span><span class="lsb">7<span class="ff77 ls8c">z<span class="ff78 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff79 lsb">17</span></span></span></span></div><div class="t m0 x37 ha y17b ff79 fs5 fc3 sc0 lsb">3<span class="ff77 ls5">y<span class="ff78 ls6">\u2212</span></span>11<span class="ff77 ls8c">z<span class="ff78 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff79 lsb">27</span></span></span></div><div class="t m0 x74 h9 y17c ff7a fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x74 h9 y17d ff7a fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x74 h9 y17e ff7a fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x75 ha y17f ff79 fs5 fc3 sc0 ls134 wsd2">11 2 9</div><div class="t m0 x75 ha y180 ff79 fs5 fc3 sc0 ls135 wsd3">02<span class="_d blank"></span><span class="ff78 lsb">\u2212<span class="ff79 lsdb">7</span>\u2212<span class="ff79">17</span></span></div><div class="t m0 x75 ha y181 ff79 fs5 fc3 sc0 lsc2 wsd4">03<span class="_15 blank"></span><span class="ff78 lsb">\u2212<span class="ff79 wsd5">11 </span>\u2212<span class="ff79">27</span></span></div><div class="t m0 x77 h9 y17c ff7a fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x77 h9 y17d ff7a fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x77 h9 y17e ff7a fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x52 ha y182 ff7b fs5 fc3 sc0 ls5">x<span class="ff7c ls4a">+</span>y<span class="ff7c ls6">+<span class="ff7d ls1">2</span></span><span class="ls8c">z<span class="ff7c lsd7">=<span class="ff7d lsb">9</span></span></span></div><div class="t m0 x37 ha y183 ff7d fs5 fc3 sc0 ls1">2<span class="ff7b ls5">y<span class="ff7c ls6">\u2212</span></span><span class="lsb">7<span class="ff7b ls8d">z<span class="ff7c ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff7d lsb">17</span></span></span></span></div><div class="t m0 x79 ha y184 ff7d fs5 fc3 sc0 lsb">3<span class="ff7b ls5">x<span class="ff7c ls8e">+</span></span>6<span class="ff7b ls5">y<span class="ff7c ls6">\u2212</span></span>5<span class="ff7b ls8d">z<span class="ff7c lsdc">=</span></span>0</div><div class="t m0 x31 h9 y185 ff7e fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x31 h9 y186 ff7e fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x31 h9 y187 ff7e fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x49 ha y182 ff7d fs5 fc3 sc0 lsdd wsd6">112 9</div><div class="t m0 x49 ha y183 ff7d fs5 fc3 sc0 ls135 wsd3">02<span class="_15 blank"></span><span class="ff7c lsb">\u2212<span class="ff7d lsdb">7</span>\u2212<span class="ff7d">17</span></span></div><div class="t m0 x49 ha y184 ff7d fs5 fc3 sc0 lsbe ws86">36<span class="_15 blank"></span><span class="ff7c lsb">\u2212<span class="ff7d ls136">50</span></span></div><div class="t m0 x6e h9 y185 ff7e fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x6e h9 y186 ff7e fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x6e h9 y187 ff7e fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xb h2a ya3 ff1 fs7 fc1 sc0 lsd ws2e">30 <span class="fsf ls7c ws7f vf">\u2022 \u2022 \u2022<span class="_12 blank"> </span></span><span class="ff3 fs10 fc2 ls7d ws80">Álgebra Linear com <span class="_0 blank"></span>Aplicações</span></div><div class="t m0 x7a h2c y188 ff2 fs10 fc2 sc0 lsb">1<span class="ff3 ls7d ws9a">.<span class="_b blank"> </span>Quais das seguintes equações são lineares em </span><span class="ff4 wsd7">x</span><span class="ff3 fs16 wsd8 v6">1</span><span class="ff3 ws7">, <span class="_4 blank"> </span></span><span class="ff4 wsd7">x<span class="ff3 fs16 lsde v6">2</span></span><span class="ff3 ws7">e </span><span class="ff4 wsd7">x</span><span class="ff3 fs16 wsd8 v6">3</span><span class="ff3">?</span></div><div class="t m0 x7a h2c y189 ff2 fs10 fc2 sc0 lsb">2<span class="ff3 ws9a">.<span class="_b blank"> </span>Sabendo que <span class="ff4 lsde">k</span><span class="ls7d">é uma constante, quais das seguintes equações são lineares?</span></span></div><div class="t m0 x7a h2c y18a ff2 fs10 fc2 sc0 lsb">3<span class="ff3 ws9a">.<span class="_b blank"> </span>Encontre o conjunto-solução de cada uma das seguintes equações lineares.</span></div><div class="t m0 x7a h2c y18b ff2 fs10 fc2 sc0 lsb">4<span class="ff3 ws9a">.<span class="_b blank"> </span>Encontre a matriz aumentada de cada um dos seguintes sistemas de equações lineares.</span></div><div class="t m0 x7a h2c y18c ff2 fs10 fc2 sc0 lsb">5<span class="ff3 ls7d ws80">.<span class="_b blank"> </span>Encontre o sistema de equações lineares correspondendo à matriz aumentada.</span></div><div class="t m0 x7a h2c y18d ff2 fs10 fc2 sc0 lsb">6<span class="ff3 ls7d ws80">.<span class="_b blank"> </span>(a) Encontre uma equação linear nas variáveis <span class="ff4 lsdf">x</span></span><span class="ff3 ws7">e <span class="_4 blank"> </span><span class="ff4 lsde">y</span><span class="ws9a">que tem <span class="ff4 lsde">x</span>= 5 + 2<span class="ff4">t</span></span>, <span class="ff4 lsde">y</span>= <span class="_4 blank"> </span><span class="ff4 lsde">t</span><span class="ws9a">como solução geral.</span></span></div><div class="t m0 x52 h2c y18e ff3 fs10 fc2 sc0 ls7d ws9a">(b)<span class="_b blank"> </span>Mostre que <span class="ff4 lse0">x</span><span class="lsb ws7">= <span class="_4 blank"> </span><span class="ff4">t</span><span class="ws9a">, <span class="_29 blank"> </span>também é a solução geral da equação da parte (a).</span></span></div><div class="t m0 x7b h45 y18f ff7f fs17 fc3 sc0 lse1">y<span class="ff80 lse2">=<span class="ff81 fs18 lsb v1">1</span></span></div><div class="t m0 x7c h46 y190 ff81 fs18 fc3 sc0 lse3">2<span class="ff7f fs17 lse4 v5">t<span class="ff80 lse5">\u2212</span></span><span class="lsb v15">5</span></div><div class="t m0 x5 h47 y190 ff81 fs18 fc3 sc0 lsb">2</div><div class="t m0 x7d h48 y191 ff82 fs17 fc3 sc0 lsb wsd9">(a) <span class="ff83 v16">\u23a1</span></div><div class="t m0 x37 h49 y192 ff83 fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x37 h49 y193 ff83 fs17 fc3 sc0 lsb">\u23a3</div><div class="t m0 x7e h4a y194 ff84 fs17 fc3 sc0 ls137">200</div><div class="t m0 x7e h4a y195 ff84 fs17 fc3 sc0 lse6">3<span class="ff85 lsb">\u2212</span><span class="ls138">40</span></div><div class="t m0 x7e h4a y196 ff84 fs17 fc3 sc0 lse7 wsda">011</div><div class="t m0 x43 h49 y197 ff83 fs17 fc3 sc0 lsb">\u23a4</div><div class="t m0 x43 h49 y192 ff83 fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x43 h4b y193 ff83 fs17 fc3 sc0 lse8">\u23a6<span class="ff82 lsb wsd9 v17">(b) </span><span class="lsb v18">\u23a1</span></div><div class="t m0 x6c h49 y192 ff83 fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x6c h49 y193 ff83 fs17 fc3 sc0 lsb">\u23a3</div><div class="t m0 x4 h4a y194 ff84 fs17 fc3 sc0 ls139 wsdb">30<span class="_2a blank"></span><span class="ff85 lsb">\u2212<span class="ff84 ls139">25</span></span></div><div class="t m0 x4 h4a y195 ff84 fs17 fc3 sc0 ls13a wsdc">714<span class="_2a blank"></span><span class="ff85 lsb">\u2212<span class="ff84">3</span></span></div><div class="t m0 x4 h4a y196 ff84 fs17 fc3 sc0 lse6">0<span class="ff85 lsb">\u2212</span><span class="lse9 wsdd">217</span></div><div class="t m0 x7f h49 y197 ff83 fs17 fc3 sc0 lsb">\u23a4</div><div class="t m0 x7f h49 y192 ff83 fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x7f h49 y193 ff83 fs17 fc3 sc0 lsb">\u23a6</div><div class="t m0 x7d h4c y198 ff82 fs17 fc3 sc0 lsb wsd9">(c) <span class="ff83 wsde v19">\ue008</span><span class="ff84 lsea wsdf v10">721<span class="_2a blank"></span><span class="ff85 lsb">\u2212<span class="ff84 ls13b">35</span></span></span></div><div class="t m0 x7e h4a y199 ff84 fs17 fc3 sc0 lseb wse0">12401</div><div class="t m0 x50 h4d y19a ff83 fs17 fc3 sc0 lsec">\ue009<span class="ff82 lsb wsd9 v1a">(d) </span><span class="lsb v1b">\u23a1</span></div><div class="t m0 x6c h49 y19b ff83 fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x6c h49 y19c ff83 fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x6c h49 y19d ff83 fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x6c h49 y19e ff83 fs17 fc3 sc0 lsb">\u23a3</div><div class="t m0 x4 h4a y19f ff84 fs17 fc3 sc0 ls13c">10007</div><div class="t m0 x4 h4a y1a0 ff84 fs17 fc3 sc0 lsed wse1">0100<span class="_2a blank"></span><span class="ff85 lsb">\u2212<span class="ff84">2</span></span></div><div class="t m0 x4 h4a y1a1 ff84 fs17 fc3 sc0 ls13c">00103</div><div class="t m0 x4 h4a y1a2 ff84 fs17 fc3 sc0 ls13d">00014</div><div class="t m0 x80 h49 y1a3 ff83 fs17 fc3 sc0 lsb">\u23a4</div><div class="t m0 x80 h49 y1a4 ff83 fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x80 h49 y1a5 ff83 fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x80 h49 y1a6 ff83 fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x80 h49 y1a7 ff83 fs17 fc3 sc0 lsb">\u23a6</div><div class="t m0 x7d h4e y1a8 ff86 fs17 fc3 sc0 lsb wse2">(a) <span class="ff87">3<span class="ff88 lsee">x</span><span class="fs18 lsef v9">1</span><span class="ff89 lsf0">\u2212</span><span class="lsf1">2<span class="ff88 lsf2">x</span><span class="fs18 lsf3 v9">2</span><span class="ff89 ls13e wse3">=\u2212<span class="_2b blank"></span><span class="ff87 lsb">1</span></span></span></span></div><div class="t m0 x81 h4a y1a9 ff87 fs17 fc3 sc0 lsf4">4<span class="ff88 lsb wse4">x</span><span class="fs18 lsf5 v9">1</span><span class="ff89 lsf0 v0">+<span class="ff87 lsb">5<span class="ff88 lsf6">x</span><span class="fs18 lsf3 v9">2</span></span><span class="lsf7">=<span class="ff87 lsb">3</span></span></span></div><div class="t m0 x81 h4a y1aa ff87 fs17 fc3 sc0 lsb">7<span class="ff88 wse4">x</span><span class="fs18 lsf5 v9">1</span><span class="ff89 lsf0 v0">+</span><span class="v0">3<span class="ff88 lsf6">x</span><span class="fs18 lsf3 v9">2</span><span class="ff89 lsf8">=</span>2</span></div><div class="t m0 x82 h4e y1ab ff86 fs17 fc3 sc0 lsb wse2">(b) <span class="ff87 lsf1">2<span class="ff88 lsf9">x</span><span class="fs18 lsfa v9">1</span><span class="ff89 lsfb v0">+</span><span class="v0">2<span class="ff88 lsb wse4">x</span><span class="fs18 lsfc v9">3</span><span class="ff89 lsfb">=</span><span class="lsb">1</span></span></span></div><div class="t m0 x50 h4a y1a9 ff87 fs17 fc3 sc0 lsb">3<span class="ff88 wse4">x</span><span class="fs18 lsfd v9">1</span><span class="ff89 lsfb v0">\u2212<span class="ff88 lsb wse4">x</span></span><span class="fs18 lsf3 v9">2</span><span class="ff89 lsfe v0">+<span class="ff87 lsf4">4<span class="ff88 lsb wse4">x</span><span class="fs18 lsfc v9">3</span></span><span class="lsfb">=</span></span><span class="v0">7</span></div><div class="t m0 x50 h4a y1aa ff87 fs17 fc3 sc0 lsb">6<span class="ff88 wse4">x</span><span class="fs18 lsfd v9">1</span><span class="ff89 lsfb v0">+<span class="ff88 lsb wse4">x</span></span><span class="fs18 lsf3 v9">2</span><span class="ff89 lsff v0">\u2212<span class="ff88 lsb wse4">x</span></span><span class="fs18 lsfc v9">3</span><span class="ff89 lsfb v0">=</span><span class="v0">0</span></div><div class="t m0 x83 h4e y1ab ff86 fs17 fc3 sc0 lsb wse2">(c) <span class="ff88 ls100">x<span class="ff87 fs18 lsfc v9">1</span><span class="ff89 lsfb v0">+<span class="ff87 lsf1">2<span class="ff88 lsb wse4">x</span><span class="fs18 ls101 v9">2</span></span><span class="ls102">\u2212<span class="ff88 lsb wse4">x<span class="ff87 fs18 ls103 v9">4</span></span></span>+<span class="ff88 lsb wse4">x<span class="ff87 fs18 lsfc v9">5</span></span>=<span class="ff87 lsb">1</span></span></span></div><div class="t m0 x84 h4a y1a9 ff87 fs17 fc3 sc0 lsb">3<span class="ff88 ls104">x</span><span class="fs18 ls105 v9">2</span><span class="ff89 lsfb v0">+<span class="ff88 lsb wse4">x</span></span><span class="fs18 ls106 v9">3</span><span class="ff89 lsfb v0">\u2212<span class="ff88 lsb wse4">x</span></span><span class="fs18 lsfc v9">5</span><span class="ff89 lsfb v0">=</span><span class="v0">2</span></div><div class="t m0 x85 h4f y1aa ff88 fs17 fc3 sc0 lsb wse4">x<span class="ff87 fs18 ls107 v9">3</span><span class="ff89 lsfb v0">+<span class="ff87 lsb">7<span class="ff88">x</span><span class="fs18 ls108 v9">4</span></span>=<span class="ff87 lsb">1</span></span></div><div class="t m0 x41 h4e y1ab ff86 fs17 fc3 sc0 lsb wse5">(d) <span class="ff88 wse4">x<span class="ff87 fs18 ls109 v9">1</span><span class="ff89 lsfb v0">=<span class="ff87 lsb">1</span></span></span></div><div class="t m0 x22 h4f y1a9 ff88 fs17 fc3 sc0 lsb wse4">x<span class="ff87 fs18 ls10a v9">2</span><span class="ff89 lsfb v0">=<span class="ff87 lsb">2</span></span></div><div class="t m0 x86 h4f y1aa ff88 fs17 fc3 sc0 ls10b">x<span class="ff87 fs18 lsfc v9">3</span><span class="ff89 lsfb v0">=<span class="ff87 lsb">3</span></span></div><div class="t m0 x7d h4e y1ac ff8a fs17 fc3 sc0 lsb wse2">(a) <span class="ff8b">7<span class="ff8c ls10c">x<span class="ff8d lsf0">\u2212</span></span>5<span class="ff8c lse1">y<span class="ff8d lsfb">=</span></span><span class="ls10d">3</span></span>(b) <span class="ff8b">3<span class="ff8c ls10e">x</span><span class="fs18 lsf5 v9">1</span><span class="ff8d lsf0">\u2212</span>5<span class="ff8c lsf6">x</span><span class="fs18 ls10f v9">2</span><span class="ff8d lsf0">+</span><span class="lsf4">4<span class="ff8c ls110">x</span><span class="fs18 ls107 v9">3</span><span class="ff8d lsfb">=</span></span>7</span></div><div class="t m0 x7d h4e y1ad ff8a fs17 fc3 sc0 lsb wse6">(c) <span class="ff8d lsf0">\u2212</span><span class="ff8b">8<span class="ff8c ls111">x</span><span class="fs18 lsf5 v9">1</span><span class="ff8d lsf0 v0">+<span class="ff8b lsf1">2<span class="ff8c ls112">x</span><span class="fs18 ls10f v9">2</span></span>\u2212</span><span class="v0">5<span class="ff8c lsf6">x</span><span class="fs18 lsef v9">3</span><span class="ff8d lsf0">+</span>6<span class="ff8c ls113">x</span><span class="fs18 ls103 v9">4</span><span class="ff8d lsfb">=</span><span class="lsec">1</span><span class="ff8a wse2">(d) </span>3<span class="ff8c ls114">v<span class="ff8d lsf0">\u2212</span></span>8<span class="ff8c ls115">w<span class="ff8d lsf0">+</span></span><span class="lsf1">2<span class="ff8c ls10c">x<span class="ff8d lsf0">\u2212</span>y<span class="ff8d lsf0">+</span></span><span class="lsf4">4<span class="ff8c ls116">z<span class="ff8d lsfb">=</span></span></span></span>0</span></span></div><div class="t m0 x7d h50 y1ae ff8e fs17 fc3 sc0 lsb wse2">(a) <span class="ff8f ls117">x<span class="ff90 fs18 lsf5 v9">1</span><span class="ff91 lsf0">\u2212</span><span class="lsf6">x<span class="ff90 fs18 ls10f v9">2</span><span class="ff91 lsf0">+</span>x<span class="ff90 fs18 ls107 v9">3</span><span class="ff91 lsfb">=</span></span></span><span class="ff90">s<span class="ff3">e</span><span class="ls118">n<span class="ff8f ls119">k</span></span></span>(b) <span class="ff8f ls13f">kx</span></div><div class="t m0 x48 h51 y1af ff90 fs18 fc3 sc0 lsef">1<span class="ff91 fs17 ls11a va">\u2212</span><span class="fs17 lsb v1c">1</span></div><div class="t m0 x87 h52 y1b0 ff8f fs17 fc3 sc0 ls11b">k<span class="ls11c v10">x</span><span class="ff90 fs18 ls11d v1d">2</span><span class="ff91 lsfb v10">=<span class="ff90 lsec">9<span class="ff8e lsb wse2">(c) </span><span class="ls11e">2</span></span></span><span class="fs18 ls11f v1e">k</span><span class="lsba v10">x</span><span class="ff90 fs18 lsf5 v1d">1</span><span class="ff91 lsf0 v10">+<span class="ff90 lsb">7<span class="ff8f lsf6">x</span><span class="fs18 ls10f v9">2</span></span>\u2212<span class="ff8f ls113">x<span class="ff90 fs18 lsfc v9">3</span></span><span class="lsfb">=<span class="ff90 lsb">0</span></span></span></div><div class="t m0 x88 h53 y1b1 ff92 fs17 fc3 sc0 lsb wse2">(a) <span class="ff93 ls117">x<span class="ff94 fs18 lsf5 v9">1</span><span class="ff95 lsf0">+</span></span><span class="ff94">5<span class="ff93 lsf6">x</span><span class="fs18 ls10f v9">2</span><span class="ff95 lsf0">\u2212<span class="ls120 v1c">\u221a</span></span><span class="lsf1">2<span class="ff93 ls121">x</span><span class="fs18 ls107 v9">3</span><span class="ff95 lsfb">=</span><span class="lsec">1</span></span></span>(b) <span class="ff93 ls122">x<span class="ff94 fs18 lsf5 v9">1</span><span class="ff95 lsf0">+</span></span><span class="ff94">3<span class="ff93 lsf6">x</span><span class="fs18 ls10f v9">2</span><span class="ff95 lsf0">+<span class="ff93 lsf6">x</span></span><span class="fs18 ls123 v9">1</span><span class="ff93 wse4">x</span><span class="fs18 lsfc v9">3</span><span class="ff95 lsfb">=</span><span class="ls124">2</span></span>(c) <span class="ff93 ls125">x<span class="ff94 fs18 ls107 v9">1</span><span class="ff95 ls13e wse3">=\u2212<span class="_2b blank"></span><span class="ff94 lsb">7<span class="ff93 wse4">x</span><span class="fs18 ls126 v9">2</span><span class="ff95 lsf0">+</span>3<span class="ff93 ls113">x</span><span class="fs18 v9">3</span></span></span></span></div><div class="t m0 x88 h54 y1b2 ff92 fs17 fc3 sc0 lsb wse2">(d) <span class="ff93 ls127">x</span><span class="ff95 fs18 vc">\u2212<span class="ff94">2</span></span></div><div class="t m0 x89 h55 y1b3 ff94 fs18 fc3 sc0 ls128">1<span class="ff95 fs17 lsf0 v7">+<span class="ff93 ls113">x</span></span><span class="ls126 v17">2</span><span class="ff95 fs17 lsf0 v7">+<span class="ff94 lsb">8<span class="ff93 ls113">x</span></span></span><span class="lsfc v17">3</span><span class="ff95 fs17 lsfb v7">=<span class="ff94 ls129">5<span class="ff92 lsb wse2">(e) <span class="ff93 ls12a">x</span></span></span></span><span class="lsb v15">3<span class="ff93">/</span>5</span></div><div class="t m0 x8a h56 y1b4 ff94 fs18 fc3 sc0 ls12b">1<span class="ff95 fs17 lsf0 v7">\u2212<span class="ff94 lsf1">2<span class="ff93 ls112">x</span></span></span><span class="ls10f v17">2</span><span class="ff95 fs17 lsf0 v7">+<span class="ff93 lsf6">x</span></span><span class="ls107 v17">3</span><span class="ff95 fs17 lsfb v7">=<span class="ff94 ls12c">4<span class="ff92 lsb wse7">(f )<span class="_b blank"> </span><span class="ff93 ls140">\u03c0x</span></span></span></span></div><div class="t m0 x14 h57 y1b5 ff94 fs18 fc3 sc0 lsf5">1<span class="ff95 fs17 lsf0 va">\u2212<span class="ls120 v1c">\u221a</span><span class="ff94 lsf1">2<span class="ff93 ls12d">x</span></span></span><span class="ls10f">2<span class="ff95 fs17 ls12e va">+</span><span class="lsb v1d">1</span></span></div><div class="t m0 x8b h58 y1b6 ff94 fs18 fc3 sc0 ls12f">3<span class="ff93 fs17 lsba v5">x</span><span class="ls107 ve">3</span><span class="ff95 fs17 lsfb v5">=<span class="ff94 lsb wse8">7<span class="fs18 v1">1<span class="ff93">/</span>3</span></span></span></div><div class="t m0 x8c he y1b7 ff1 fs7 fc6 sc0 lsbb wsea">Conjunto de Exercícios 1.1</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y25 w4 h59" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg5.png"><div class="t m0 x7a h5a y1b8 ff2 fs10 fc2 sc0 lsb">7<span class="ff3 ls7d ws7">.<span class="_b blank"> </span>A<span class="_13 blank"> </span>curva <span class="ff4 ls141">y</span><span class="lsb">= <span class="_4 blank"> </span></span></span><span class="ff4 wsd7">ax<span class="ff3 fs16 ls142 v5">2</span></span><span class="ff3 ws7">+ </span><span class="ff4 wseb">bx </span><span class="ff3 ws7">+ <span class="ff4 ls141">c</span><span class="ls7d wsf4">mostrada na figura passa pelos pontos (<span class="ff4 ls143">x</span></span><span class="fs16 wsd8 v6">1</span>, <span class="ff4 ls144">y</span><span class="fs16 wsd8 v6">1</span><span class="ls7c wsf4">), (<span class="ff4 ls145">x</span></span><span class="fs16 wsd8 v6">2</span>, <span class="ff4 ls144">y</span><span class="fs16 wsd8 v6">2</span><span class="ls7d wsf5">) e (</span></span><span class="ff4 wsd7">x</span><span class="ff3 fs16 wsd8 v6">3</span><span class="ff3 ws7">, <span class="_4 blank"> </span><span class="ff4 ls144">y</span><span class="fs16 wsd8 v6">3</span><span class="ls7d wsf4">). Mostre que os coeficientes </span><span class="ff4">a</span>, <span class="ff4 ls141">b</span>e <span class="ff4 ls141">c</span><span class="ls7d wsf4">são uma solução</span></span></div><div class="t m0 x13 h2c y1b9 ff3 fs10 fc2 sc0 lsb ws9a">do sistema de equações lineares cuja matriz aumentada é</div><div class="t m0 x8d h2c y1ba ff2 fs10 fc2 sc0 lsb">8<span class="ff3 ls7d ws9a">.<span class="_b blank"> </span>Considere o sistema de equações</span></div><div class="t m0 x13 h2c y1bb ff3 fs10 fc2 sc0 ls7d ws9a">Mostre que para este sistema ser consistente, as constantes <span class="ff4 lsb">a<span class="ff3 ws7">, </span><span class="lsde">b</span><span class="ff3 ws7">e <span class="_4 blank"> </span></span><span class="lsde">c</span></span>devem satisfazer <span class="ff4 ls146">c</span><span class="lsb ws7">= <span class="_4 blank"> </span><span class="ff4 lsde">a</span>+ <span class="ff4">b</span>.</span></div><div class="t m0 x8d h5b y1bc ff2 fs10 fc2 sc0 lsb">9<span class="ff3 ls7d ws9a">.<span class="_b blank"> </span>Mostre que se as equações lineares <span class="ff4 ls35">x</span><span class="fs16 lsde v8">1</span></span><span class="ff3 ws7 v0">+ <span class="_4 blank"> </span><span class="ff4 wsd7">kx</span><span class="fs16 lsde v6">2</span>= <span class="ff4 lsde">c</span>e <span class="_4 blank"> </span><span class="ff4 wsd7">x</span><span class="fs16 lsde v6">1</span>+ <span class="_4 blank"> </span><span class="ff4 wsd7">lx</span><span class="fs16 lsde v6">2</span>= <span class="ff4 lsde">d</span><span class="ws9a">têm o mesmo conjunto-solução, então as equações são idênticas.</span></span></div><div class="t m0 xb he y1bd ff1 fs7 fc1 sc0 ls71 wsea">Discussão e Descoberta</div><div class="t m0 xb h2c y1be ff2 fs10 fc2 sc0 lsb">10<span class="ff3 ls7d ws9a">.<span class="_b blank"> </span>Para quais valores da constante <span class="ff4 lsde">k</span>o sistema</span></div><div class="t m0 x13 h2c y1bf ff3 fs10 fc2 sc0 lsb ws9a">não tem solução? Exatamente uma solução? Infinitas soluções? Explique seu raciocínio.</div><div class="t m0 x1 h2c y1c0 ff2 fs10 fc2 sc0 ls159 wsec">11 <span class="ff3 ls7d ws9a">.<span class="_b blank"> </span>Considere o sistema de equações</span></div><div class="t m0 x13 h2c y1c1 ff3 fs10 fc2 sc0 ls7d ws9a">O que você pode dizer sobre a posição relativa das retas <span class="ff4 lsb">ax <span class="ff3 ws7">+ </span>by <span class="ff3 ws7">= <span class="_4 blank"> </span></span>k<span class="ff3 ws7">, <span class="_4 blank"> </span></span>cx <span class="ff3 ws7">+ </span>dy <span class="ff3 ws7">= <span class="_4 blank"> </span></span><span class="lsde">l</span><span class="ff3 ws7">e <span class="_4 blank"> </span></span>ex <span class="ff3 ws7">+ <span class="_4 blank"> </span></span>fy <span class="ff3 ws7">= </span>m<span class="ff3">, quando</span></span></div><div class="t m0 x52 h2c y1c2 ff3 fs10 fc2 sc0 lsb ws9a">(a)<span class="_b blank"> </span>o sistema não tem solução;</div><div class="t m0 x13 h2c y1c3 ff3 fs10 fc2 sc0 lsb ws9a">(b)<span class="_b blank"> </span>o sistema tem exatamente uma solução;</div><div class="t m0 x52 h2c y1c4 ff3 fs10 fc2 sc0 ls7d ws9a">(c)<span class="_b blank"> </span>o sistema tem infinitas soluções?</div><div class="t m0 xb h2c y1c5 ff2 fs10 fc2 sc0 lsb">12<span class="ff3 ls7d wsf6">.<span class="_b blank"> </span>Se o sistema do Exercício 11 for consistente, explique por que pelo menos uma das equações poderá ser descartada do sistema sem alterar o</span></div><div class="t m0 x13 h2c y1c6 ff3 fs10 fc2 sc0 lsb">conjunto-solução.</div><div class="t m0 xb h2c y1c7 ff2 fs10 fc2 sc0 lsb">13<span class="ff3 ls7d ws7">.<span class="_b blank"> </span>Se <span class="_4 blank"> </span><span class="ff4 ls147">k</span><span class="lsb">= <span class="ff4 ls147">l</span>= <span class="_4 blank"> </span><span class="ff4 ls147">m</span></span><span class="wsf7">no Exercício 11, explique por que o sistema deve ser consistente. O que pode ser dito sobre o ponto de corte das três retas se <span class="lsb">o</span></span></span></div><div class="t m0 x13 h2c y1c8 ff3 fs10 fc2 sc0 lsb ws9a">sistema tem exatamente uma solução?</div><div class="t m0 x8e h5c y1c9 ff96 fs17 fc3 sc0 ls15a wsed">ax <span class="ff97 ls148">+</span><span class="ls15b wsee">by <span class="ff97 lsfb">=</span><span class="lsb">k</span></span></div><div class="t m0 x8e h5c y1ca ff96 fs17 fc3 sc0 ls15c wsef">cx <span class="ff97 ls149">+</span><span class="ls15d wsf0">dy <span class="ff97 lsfb">=</span><span class="lsb">l</span></span></div><div class="t m0 x8e h5c y1cb ff96 fs17 fc3 sc0 ls15c wsef">ex <span class="ff97 lsfb">+</span><span class="ls15e wsf1">fy <span class="ff97 lsfb">=</span><span class="lsb">m</span></span></div><div class="t m0 x8f h4a y1cc ff98 fs17 fc3 sc0 lse1">x<span class="ff99 lsff">\u2212</span>y<span class="ff99 lsfb">=<span class="ff9a lsb">3</span></span></div><div class="t m0 x8e h4a y1cd ff9a fs17 fc3 sc0 lsf1">2<span class="ff98 lse1">x<span class="ff99 lsfb">\u2212</span></span>2<span class="ff98 lse1">y<span class="ff99 lsfb">=</span><span class="lsb">k</span></span></div><div class="t m0 x8f h4a y1ce ff9b fs17 fc3 sc0 lse1">x<span class="ff9c lsfb">+</span>y<span class="ff9c lsfb">+<span class="ff9d lsf1">2</span></span><span class="ls116">z<span class="ff9c lsfb">=</span><span class="lsb">a</span></span></div><div class="t m0 x8f h5c y1cf ff9b fs17 fc3 sc0 ls14a">x<span class="ff9c ls14b">+</span><span class="ls116">z<span class="ff9c lsfb">=</span><span class="lsb">b</span></span></div><div class="t m0 x8e h4a y1d0 ff9d fs17 fc3 sc0 lsf1">2<span class="ff9b lse1">x<span class="ff9c lsfb">+</span>y<span class="ff9c ls14c">+</span></span><span class="lsb">3<span class="ff9b ls116">z<span class="ff9c lsfb">=</span><span class="lsb">c</span></span></span></div><div class="t m0 x52 h49 y1d1 ff9e fs17 fc3 sc0 lsb">\u23a1</div><div class="t m0 x52 h49 y1d2 ff9e fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x52 h49 y1d3 ff9e fs17 fc3 sc0 lsb">\u23a2</div><div class="t m0 x52 h49 y1d4 ff9e fs17 fc3 sc0 lsb">\u23a3</div><div class="t m0 x8f h5d y1d5 ff9f fs17 fc3 sc0 ls14d">x<span class="ffa0 fs18 lsb v1">2</span></div><div class="t m0 x90 h5e y1d6 ffa0 fs18 fc3 sc0 ls14e">1<span class="ff9f fs17 lsb wsde v7">x</span><span class="ls14f v17">1</span><span class="fs17 ls150 v7">1<span class="ff9f ls151">y</span></span><span class="lsb v17">1</span></div><div class="t m0 x8f h5f y1d7 ff9f fs17 fc3 sc0 ls152">x<span class="ffa0 fs18 lsb v1">2</span></div><div class="t m0 x90 h5e y1d8 ffa0 fs18 fc3 sc0 ls14e">2<span class="ff9f fs17 lsb wsde v7">x</span><span class="ls14f v17">2</span><span class="fs17 ls150 v7">1<span class="ff9f ls151">y</span></span><span class="lsb v17">2</span></div><div class="t m0 x8f h5f y1d9 ff9f fs17 fc3 sc0 ls152">x<span class="ffa0 fs18 lsb v1">2</span></div><div class="t m0 x90 h5e y1da ffa0 fs18 fc3 sc0 ls14e">3<span class="ff9f fs17 lsb wsde v7">x</span><span class="ls14f v17">3</span><span class="fs17 ls150 v7">1<span class="ff9f ls151">y</span></span><span class="lsb v17">3</span></div><div class="t m0 x72 h49 y1d1 ff9e fs17 fc3 sc0 lsb">\u23a4</div><div class="t m0 x72 h49 y1d2 ff9e fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x72 h49 y1d3 ff9e fs17 fc3 sc0 lsb">\u23a5</div><div class="t m0 x72 h49 y1d4 ff9e fs17 fc3 sc0 lsb">\u23a6</div><div class="t m0 x6c h60 y1db ffa1 fs11 fc3 sc0 lsb">y</div><div class="t m0 x91 h60 y1dc ffa1 fs11 fc3 sc0 lsb">x</div><div class="t m0 x92 h61 y1dd ffa1 fs11 fc3 sc0 lsb">y<span class="ffa2 ws7"> = </span><span class="wsf2">ax</span><span class="ffa2 fs19 wsf3 v5">2</span><span class="ffa2 ws7"> + </span>bx<span class="ffa2 ws7"> + </span>c</div><div class="t m0 x75 h62 y1de ffa2 fs11 fc3 sc0 lsb">(<span class="ffa1 ls153">x</span><span class="fs19 wsf3 v8">1</span><span class="ls15f ws7">, <span class="_4 blank"> </span><span class="ffa1 ls25">y</span></span><span class="fs19 wsf3 v8">1</span>)</div><div class="t m0 x70 h62 y1df ffa2 fs11 fc3 sc0 lsb">(<span class="ffa1 ls35">x</span><span class="fs19 wsf3 v8">3</span><span class="ls15f ws7">, <span class="_4 blank"> </span><span class="ffa1 ls25">y</span></span><span class="fs19 wsf3 v8">3</span>)</div><div class="t m0 x93 h63 y1e0 ffa2 fs11 fc3 sc0 lsb">(<span class="ffa1 wsf2">x</span><span class="fs19 wsf3 v8">2</span><span class="ls15f ws7 v0">, <span class="_4 blank"> </span><span class="ffa1 ls25">y</span><span class="fs19 ls154 v8">2</span><span class="lsb">)</span></span></div><div class="t m0 x80 h64 y1e1 ffa3 fs17 fc1 sc0 ls160">Figur<span class="ffa4 ls155">a</span><span class="lsb wsde">Ex-<span class="_2b blank"></span>7</span></div><div class="t m0 x3f h38 ya3 ff4 fs10 fc2 sc0 ls7d ws9a">Capítulo 1 - Sistemas de Equações Linear<span class="_0 blank"></span>es e Matrizes<span class="_12 blank"> </span><span class="ff1 fsf fc1 ls7c ws7f v14">\u2022 \u2022 \u2022<span class="_12 blank"> </span><span class="fs7 lsd v5">31</span></span></div><div class="t m0 x1 h10 y1e2 ff1 fs8 fc1 sc0 ls6f ws14">1.2 <span class="fs9 fc2 ls8 ws2f v5">ELIMINAÇÃO GAUSSIANA</span></div><div class="t m0 xb h7 y1e3 ffc fs2 fc5 sc0 lsa wsf8">Nesta seção nós vamos desenvolver um procedimento sistemáti-</div><div class="t m0 xb h7 y1e4 ffc fs2 fc5 sc0 lsa wsf9">co para resolver sistemas de equações lineares. O procedimento</div><div class="t m0 xb h7 y1e5 ffc fs2 fc5 sc0 lsa wsfa">é baseado na idéia de reduzir a matriz aumentada de um sistema</div><div class="t m0 xb h7 y1e6 ffc fs2 fc5 sc0 lsa wsfb">a uma outra matriz aumentada que seja suficientemente simples</div><div class="t m0 xb h7 y1e7 ffc fs2 fc5 sc0 lsa ws34">a ponto de permitir visualizar a solução.</div><div class="t m0 xb he y1e8 ff1 fs7 fc1 sc0 ls161 wsfc">Forma Escalonada<span class="_12 blank"> </span><span class="ff3 fs2 fc2 lsb wsc0">No Exemplo 3 da última seção nós</span></div><div class="t m0 xb h4 y1e9 ff3 fs2 fc2 sc0 lsa wsfd">resolvemos um sistema linear nas incógnitas <span class="ff4 lsb">x</span><span class="ls156">,<span class="ff4">y</span>e<span class="ff4">z</span><span class="lsb">reduzindo</span></span></div><div class="t m0 xb h4 y1ea ff3 fs2 fc2 sc0 lsb ws3c">a matriz aumentada à forma</div><div class="t m0 xb h4 y1eb ff3 fs2 fc2 sc0 lsb wsfe">a partir da qual a solução <span class="ff4 ls157">x</span>= 1, <span class="ff4 ls158">y</span>= 2, <span class="ff4 ls158">z</span>= 3 ficou evidente. Isto</div><div class="t m0 xb h4 y1ec ff3 fs2 fc2 sc0 lsb wsff">é um exemplo de uma matriz que está em <span class="ffd lsa">forma escalonada</span></div><div class="t m0 x14 h4 y1ed ffd fs2 fc2 sc0 lsa ws100">reduzida por linhas<span class="ff3 ws101">. Para ser desta forma, uma matriz deve ter</span></div><div class="t m0 x14 h4 y1ee ff3 fs2 fc2 sc0 lsa ws6">as seguintes propriedades:</div><div class="t m0 x15 h4 y1ef ff2 fs2 fc1 sc0 lsb ws84">1. <span class="ff3 fc2 lsa ws102">Se uma linha não consistir só de zeros, então o primeiro</span></div><div class="t m0 x3a h4 y1f0 ff3 fs2 fc2 sc0 lsb ws103">número não-nulo da linha é um 1. Chamamos este</div><div class="t m0 x3a h4 y1f1 ff3 fs2 fc2 sc0 lsb ws3c">número 1 de <span class="ffd lsa">líder </span><span class="ws7">ou <span class="_4 blank"> </span><span class="ffd">pivô</span>.</span></div><div class="t m0 x15 h4 y1f2 ff2 fs2 fc1 sc0 lsb ws84">2. <span class="ff3 fc2 lsa ws104">Se existirem linhas constituídas somente de zeros, elas</span></div><div class="t m0 x3a h4 y1f3 ff3 fs2 fc2 sc0 lsa ws6">estão agrupadas juntas nas linhas inferiores da matriz.</div><div class="t m0 x15 h4 y1f4 ff2 fs2 fc1 sc0 lsb ws84">3. <span class="ff3 fc2 lsa ws105">Em quaisquer duas linhas sucessivas que não consistem</span></div><div class="t m0 x3a h4 y1f5 ff3 fs2 fc2 sc0 lsa ws106">só de zeros, o líder da linha inferior ocorre mais à di-</div><div class="t m0 x3a h4 y1f6 ff3 fs2 fc2 sc0 lsb ws3c">reita que o líder da linha superior<span class="_0 blank"></span>.</div><div class="t m0 x15 h4 y1f7 ff2 fs2 fc1 sc0 lsb ws84">4. <span class="ff3 fc2 ws107">Cada coluna que contém um líder tem zeros nas demais</span></div><div class="t m0 x3a h4 y1f8 ff3 fs2 fc2 sc0 lsa">entradas.</div><div class="t m0 x14 h4 y1f9 ff3 fs2 fc2 sc0 lsa ws108">Dizemos que uma matriz que tem as três primeiras propriedades</div><div class="t m0 x14 h4 y1fa ff3 fs2 fc2 sc0 lsa ws109">está em <span class="ffd ws10a">forma escalonada por linhas<span class="ff3 lsb">, ou simplesmente, em</span></span></div><div class="t m0 x14 h4 y1fb ffd fs2 fc2 sc0 lsa ws10b">forma escalonada<span class="ff3">. (Assim, uma matriz em forma escalonada</span></div><div class="t m0 x14 h4 y1fc ff3 fs2 fc2 sc0 lsa ws10c">reduzida por linhas necessariamente está em forma escalonada,</div><div class="t m0 x14 h4 y1fd ff3 fs2 fc2 sc0 lsa ws6">mas não reciprocamente.)</div><div class="t m0 x7b h9 y1fe ffa5 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x7b h9 y1ff ffa5 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x4b ha y200 ffa6 fs5 fc3 sc0 ls162">1001</div><div class="t m0 x4b ha y201 ffa6 fs5 fc3 sc0 ls162">0102</div><div class="t m0 x4b ha y202 ffa6 fs5 fc3 sc0 ls162">0013</div><div class="t m0 x94 h9 y1fe ffa5 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x94 h9 y1ff ffa5 fs5 fc3 sc0 lsb">\u23a6</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y25 w9 hf" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg6.png"><div class="t m0 xb h4 y203 ff3 fs2 fc2 sc0 lsa ws12e">As seguintes matrizes estão em forma escalonada reduzida por</div><div class="t m0 xb h4 y204 ff3 fs2 fc2 sc0 lsa">linhas.</div><div class="t m0 xb h4 y205 ff3 fs2 fc2 sc0 lsa ws6">As seguintes matrizes estão em forma escalonada.</div><div class="t m0 xb h4 y206 ff3 fs2 fc2 sc0 lsa ws12f">Nós deixamos para você a tarefa de confirmar que cada uma das</div><div class="t m0 xb h14 y207 ff3 fs2 fc2 sc0 lsa ws6">matrizes deste exemplo satisfaz todos os requisitos exigidos.<span class="_3 blank"> </span><span class="ffe fc1 lsb">®</span></div><div class="t m0 xb h4 y208 ff3 fs2 fc2 sc0 lsb ws130">Como o último exemplo ilustra, uma matriz em forma escalo-</div><div class="t m0 xb h4 y209 ff3 fs2 fc2 sc0 lsb ws131">nada tem zeros abaixo de cada líder<span class="_0 blank"></span>, enquanto que uma matriz</div><div class="t m0 xb h4 y20a ff3 fs2 fc2 sc0 lsb ws132">em forma escalonada reduzida por linhas tem zeros abaixo <span class="ff4">e</span></div><div class="t m0 xb h4 y20b ff4 fs2 fc2 sc0 lsa ws10d">acima <span class="ff3 lsb ws133">de cada líder<span class="_0 blank"></span>. <span class="_0 blank"></span>Assim, colocando qualquer número real no</span></div><div class="t m0 xb h4 y20c ff3 fs2 fc2 sc0 lsa ws134">lugar dos asteriscos, as matrizes dos seguintes tipos estão em</div><div class="t m0 xb h4 y20d ff3 fs2 fc2 sc0 lsa ws6">forma escalonada:</div><div class="t m0 xb h4 y20e ff3 fs2 fc2 sc0 lsa ws135">Além disto, as matrizes dos seguintes tipos estão em forma</div><div class="t m0 xb h4 y20f ff3 fs2 fc2 sc0 lsb ws3c">escalonada reduzida por linhas:</div><div class="t m0 x95 h14 y210 ffe fs2 fc1 sc0 lsb">®</div><div class="t m0 x13 h4 y211 ff3 fs2 fc2 sc0 lsb ws136">Se a matriz aumentada de um sistema de equações lineares</div><div class="t m0 xb h4 y212 ff3 fs2 fc2 sc0 lsa ws137">for colocada em forma escalonada reduzida por linhas por meio</div><div class="t m0 xb h4 y213 ff3 fs2 fc2 sc0 lsb ws1c">de uma seqüência de operações elementares, então a solução do</div><div class="t m0 xb h4 y214 ff3 fs2 fc2 sc0 lsb ws138">sistema está visível ou então se torna visível depois de uns</div><div class="t m0 xb h4 y215 ff3 fs2 fc2 sc0 lsa ws6">poucos passos simples. O próximo exemplo ilustra isto.</div><div class="t m0 x14 h4 y216 ff3 fs2 fc2 sc0 lsb ws139">Suponha que a matriz aumentada de um sistema de equações</div><div class="t m0 x14 h4 y217 ff3 fs2 fc2 sc0 ls19c ws13a">lineares foi reduzida por operações sobre linhas à forma</div><div class="t m0 x14 h4 y218 ff3 fs2 fc2 sc0 lsb ws3c">escalonada reduzida por linhas dada. Resolva o sistema.</div><div class="t m0 x14 h4 y219 ff4 fs2 fc1 sc0 lsb ws6">Solução <span class="ff3">(</span>a<span class="ff3 ls72 ws10e">). <span class="fc2 lsa ws6">O sistema de equações correspondente é</span></span></div><div class="t m0 x14 h65 y21a ff3 fs2 fc2 sc0 lsa ws6">Por inspeção, <span class="ff4 ls163">x</span><span class="fs3 lsb ws7 v6">1 </span><span class="lsb v0">= 5, <span class="ff4 ws17">x</span><span class="fs3 ls164 v6">2</span>= \u20132, <span class="ff4 ws17">x</span><span class="fs3 ls2c v6">3</span>= 4.</span></div><div class="t m0 x14 h4 y21b ff4 fs2 fc1 sc0 lsb ws6">Solução <span class="ff3">(</span>b<span class="ff3 ls72 ws10e">). <span class="fc2 lsa ws6">O sistema de equações correspondente é</span></span></div><div class="t m0 x14 h65 y21c ff3 fs2 fc2 sc0 lsb ws7">Como <span class="_a blank"> </span><span class="ff4 ws17">x</span><span class="fs3 ws16 v6">1</span><span class="ls10 v0">,<span class="ff4 lsb ws17">x</span><span class="fs3 ls165 v6">2</span><span class="ls166">e<span class="ff4 lsb ws17">x</span><span class="fs3 ls165 v6">3</span><span class="lsb ws13b">correspondem a líderes na matriz aumentada,</span></span></span></div><div class="t m0 x14 h4 y21d ff3 fs2 fc2 sc0 lsa ws13c">dizemos que estas variáveis são <span class="ffd">variáveis líderes</span><span class="ws13d">. As <span class="_4 blank"> </span>variáveis</span></div><div class="t m0 x14 h4 y21e ff3 fs2 fc2 sc0 lsa ws13e">não-líderes (neste caso, só <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">4</span></span>) são chamadas <span class="ffd">variáveis livres</span><span class="lsb">.</span></div><div class="t m0 x14 h4 y21f ff3 fs2 fc2 sc0 lsa ws101">Resolvendo as variáveis líderes em termos das variáveis livres,</div><div class="t m0 x14 h4 y220 ff3 fs2 fc2 sc0 lsa">obtemos</div><div class="t m0 x14 h4 y221 ff3 fs2 fc2 sc0 lsa ws13f">A<span class="_6 blank"> </span>partir deste formato das equações nós vemos que podemos dar</div><div class="t m0 x14 h66 y222 ff3 fs2 fc2 sc0 lsb ws140">um valor arbitrário à variável livre <span class="ff4 ws17">x</span><span class="fs3 ws16 v6">4</span><span class="v0">, digamos <span class="ff4">t</span>, que então</span></div><div class="t m0 x14 h4 y223 ff3 fs2 fc2 sc0 lsa ws141">determina os valores das variáveis líderes <span class="ff4 ls167">x</span><span class="fs3 lsb ws16 v6">1</span><span class="ls168">,<span class="ff4 lsb ws17">x</span><span class="fs3 ls169 v6">2</span><span class="ls16a">e<span class="ff4 ls16b">x</span><span class="fs3 lsb ws16 v6">3</span><span class="lsb">. Resulta,</span></span></span></div><div class="t m0 x14 h4 y224 ff3 fs2 fc2 sc0 lsb ws142">assim, que há uma infinidade de soluções e que a solução geral</div><div class="t m0 x14 h4 y225 ff3 fs2 fc2 sc0 lsb ws3c">é dada pela fórmula</div><div class="t m0 x14 h4 y226 ff4 fs2 fc1 sc0 lsb ws10f">Solução <span class="ff3">(</span>c<span class="ff3 ls72 ws110">). </span><span class="ff3 fc2">A<span class="_14 blank"> </span>linha de zeros leva à equação 0<span class="ff4 ws17">x</span><span class="fs3 ls16c v6">1</span><span class="v0">+ 0<span class="ff4 ws17">x</span><span class="fs3 ls16c v6">2</span>+ 0<span class="ff4 ws17">x</span><span class="fs3 ls16c v6">3</span>+</span></span></div><div class="t m0 x14 h4 y227 ff3 fs2 fc2 sc0 lsb">0<span class="ff4 ls16b">x</span><span class="fs3 ws7 v6">4 </span><span class="ws143">+ 0</span><span class="ff4 ws17">x</span><span class="fs3 ls16d v6">5</span><span class="lsa ws144">= 0, que não coloca restrições às soluções (por quê?).</span></div><div class="t m0 x14 h4 y228 ff3 fs2 fc2 sc0 lsa ws145">Assim, podemos omitir esta equação e escrever o sistema cor-</div><div class="t m0 x14 h4 y229 ff3 fs2 fc2 sc0 lsa ws6">respondente como</div><div class="t m0 x14 h4 y22a ff3 fs2 fc2 sc0 lsa ws146">Aqui, as variáveis líderes são <span class="ff4 ls16e">x</span><span class="fs3 lsb ws16 v6">1</span><span class="ls16f v0">,<span class="ff4 ls170">x</span><span class="fs3 ls171 v6">3</span><span class="lsb ws7">e <span class="ff4 ws17">x</span><span class="fs3 ls171 v6">4</span></span><span class="lsa">e as variáveis livres são</span></span></div><div class="t m0 x14 h4 y22b ff4 fs2 fc2 sc0 ls163">x<span class="ff3 fs3 ls172 v6">2</span><span class="ff3 ls173">e</span><span class="lsb ws17">x<span class="ff3 fs3 ws16 v6">5</span></span><span class="ff3 lsa ws147">. Resolvendo as variáveis líderes em termos das variáveis</span></div><div class="t m0 x14 h4 y22c ff3 fs2 fc2 sc0 lsa ws6">livres obtemos</div><div class="t m0 x96 ha y22d ffa7 fs5 fc3 sc0 lsb ws24">x<span class="ffa8 fsc ls174 v9">1</span><span class="ffa9 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ffa8 ls8e">2<span class="ffa9 ls6">\u2212</span><span class="lsb">6<span class="ffa7 ws24">x</span><span class="fsc ls175 v9">2</span><span class="ffa9 ls4">\u2212</span><span class="ls2">4<span class="ffa7 ls176">x</span></span><span class="fsc v9">5</span></span></span></span></div><div class="t m0 x96 ha y22e ffa7 fs5 fc3 sc0 lsb ws24">x<span class="ffa8 fsc ls174 v9">3</span><span class="ffa9 ls177">=<span class="ffa8 ls6">1<span class="ffa9">\u2212</span><span class="lsb">3</span></span></span>x<span class="ffa8 fsc v9">5</span></div><div class="t m0 x96 ha y22f ffa7 fs5 fc3 sc0 lsb ws24">x<span class="ffa8 fsc ls174 v9">4</span><span class="ffa9 ls178">=<span class="ffa8 ls8e">2</span><span class="ls6">\u2212</span></span><span class="ffa8">5</span>x<span class="ffa8 fsc v9">5</span></div><div class="t m0 x97 h67 y230 ffaa fs5 fc3 sc0 lsb ws24">x<span class="ffab fsc ls4d v9">1</span><span class="ffac ls4 v0">+<span class="ffab lsb">6<span class="ffaa ls50">x</span><span class="fsc ls179 v9">2</span></span><span class="ls6">+<span class="ffab ls2">4</span></span></span><span class="v0">x<span class="ffab fsc ls51 v9">5</span><span class="ffac ls75 ws25">=\u2212<span class="_d blank"></span><span class="ffab lsb">2</span></span></span></div><div class="t m0 x98 h68 y231 ffaa fs5 fc3 sc0 lsb ws24">x<span class="ffab fsc ls17a v9">3</span><span class="ffac ls6 v0">+<span class="ffab lsb">3<span class="ffaa">x</span><span class="fsc ls17b v9">5</span></span><span class="ls177">=<span class="ffab lsb">1</span></span></span></div><div class="t m0 x99 h1a y232 ffaa fs5 fc3 sc0 lsb ws24">x<span class="ffab fsc ls174 v9">4</span><span class="ffac ls6 v0">+<span class="ffab lsb">5<span class="ffaa">x</span><span class="fsc ls17b v9">5</span></span><span class="ls178">=<span class="ffab lsb">2</span></span></span></div><div class="t m0 x9a ha y233 ffad fs5 fc3 sc0 ls36">x<span class="ffae fsc ls53 v9">1</span><span class="ffaf ls75 ws25">=\u2212<span class="_d blank"></span><span class="ffae ls4">1<span class="ffaf">\u2212</span><span class="ls2">4<span class="ffad ls54 ws26">t, x</span></span></span></span></div><div class="t m0 x9b h69 y234 ffae fsc fc3 sc0 ls55">2<span class="ffaf fs5 ls6 va">=<span class="ffae ls4">6<span class="ffaf">\u2212</span><span class="ls1">2<span class="ffad ls54 ws26">t, x</span></span></span></span></div><div class="t m0 x9c h69 y234 ffae fsc fc3 sc0 ls53">3<span class="ffaf fs5 ls6 va">=<span class="ffae ls17c">2<span class="ffaf ls4">\u2212</span><span class="lsb">3<span class="ffad ls54 ws26">t, x</span></span></span></span></div><div class="t m0 x9d h6a y234 ffae fsc fc3 sc0 ls6a">4<span class="ffaf fs5 ls6 va">=<span class="ffad lsb">t</span></span></div><div class="t m0 x9e h67 y235 ffb0 fs5 fc3 sc0 lsb ws24">x<span class="ffb1 fsc ls55 v9">1</span><span class="ffb2 ls75 ws25 v0">=\u2212<span class="_d blank"></span><span class="ffb1 ls6">1<span class="ffb2">\u2212</span><span class="ls2">4<span class="ffb0 lsb ws24">x<span class="ffb1 fsc v9">4</span></span></span></span></span></div><div class="t m0 x9e ha y236 ffb0 fs5 fc3 sc0 lsb ws24">x<span class="ffb1 fsc ls55 v9">2</span><span class="ffb2 ls178">=<span class="ffb1 ls6">6<span class="ffb2">\u2212</span><span class="ls1">2</span></span></span>x<span class="ffb1 fsc v9">4</span></div><div class="t m0 x9e ha y237 ffb0 fs5 fc3 sc0 lsb ws24">x<span class="ffb1 fsc ls55 v9">3</span><span class="ffb2 ls17d">=<span class="ffb1 ls8e">2</span><span class="ls6">\u2212</span></span><span class="ffb1">3</span>x<span class="ffb1 fsc v9">4</span></div><div class="t m0 x9f h6b y238 ffb3 fs5 fc3 sc0 lsb ws24">x<span class="ffb4 fsc ls17e v9">1</span><span class="ffb5 ls6 v0">+<span class="ffb4 ls2">4</span></span><span class="v0">x<span class="ffb4 fsc ls17f v9">4</span><span class="ffb5 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ffb4 lsb">1</span></span></span></div><div class="t m0 x29 h68 y239 ffb3 fs5 fc3 sc0 ls180">x<span class="ffb4 fsc ls181 v9">2</span><span class="ffb5 ls6 v0">+<span class="ffb4 ls1">2<span class="ffb3 lsb ws24">x</span><span class="fsc ls174 v9">4</span></span><span class="ls178">=<span class="ffb4 lsb">6</span></span></span></div><div class="t m0 xa0 h2f y23a ffb3 fs5 fc3 sc0 lsb ws24">x<span class="ffb4 fsc ls53 v9">3</span><span class="ffb5 ls6 v0">+<span class="ffb4 lsb">3<span class="ffb3">x</span><span class="fsc ls182 v9">4</span></span><span class="ls17d">=<span class="ffb4 lsb">2</span></span></span></div><div class="t m0 x9b ha y23b ffb6 fs5 fc3 sc0 lsb ws24">x<span class="ffb7 fsc ls17e v9">1</span><span class="ffb8 ls177">=</span><span class="ffb7">5</span></div><div class="t m0 x98 h1a y23c ffb6 fs5 fc3 sc0 lsb ws24">x<span class="ffb7 fsc ls181 v9">2</span><span class="ffb8 ls75 wscf v0">=\u2212<span class="_d blank"></span><span class="ffb7 lsb">2</span></span></div><div class="t m0 x99 h1a y23d ffb6 fs5 fc3 sc0 lsb ws24">x<span class="ffb7 fsc ls53 v9">3</span><span class="ffb8 ls183 v0">=<span class="ffb7 lsb">4</span></span></div><div class="t m0 xa1 h6c y23e ffb9 fs5 fc3 sc0 lsb ws111">(a) <span class="ffba v1f">\u23a1</span></div><div class="t m0 x3b h9 y23f ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3b h9 y240 ffba fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x5a ha y241 ffbb fs5 fc3 sc0 ls134 ws112">1005</div><div class="t m0 x5a ha y242 ffbb fs5 fc3 sc0 ls130 ws113">010<span class="_15 blank"></span><span class="ffbc lsb">\u2212<span class="ffbb">2</span></span></div><div class="t m0 x5a ha y243 ffbb fs5 fc3 sc0 ls184 ws114">0014</div><div class="t m0 xa2 h9 y244 ffba fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa2 h9 y23f ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa2 h6d y240 ffba fs5 fc3 sc0 ls185">\u23a6<span class="ffb9 lsb ws111 v17">(b) </span><span class="lsb v4">\u23a1</span></div><div class="t m0 x25 h9 y23f ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x25 h9 y240 ffba fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x3d ha y241 ffbb fs5 fc3 sc0 lsc2 wsd4">1004<span class="_15 blank"></span><span class="ffbc lsb">\u2212<span class="ffbb">1</span></span></div><div class="t m0 x3d ha y242 ffbb fs5 fc3 sc0 ls130 ws113">01026</div><div class="t m0 x3d ha y243 ffbb fs5 fc3 sc0 ls186 ws115">00132</div><div class="t m0 xa3 h9 y244 ffba fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa3 h9 y23f ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa3 h9 y240 ffba fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa1 h6e y245 ffb9 fs5 fc3 sc0 lsb">(c)</div><div class="t m0 x3b h9 y246 ffba fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x3b h9 y247 ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3b h9 y248 ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3b h9 y249 ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3b h9 y24a ffba fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x5a ha y24b ffbb fs5 fc3 sc0 lsc2 wsd4">16004<span class="_15 blank"></span><span class="ffbc lsb">\u2212<span class="ffbb">2</span></span></div><div class="t m0 x5a ha y24c ffbb fs5 fc3 sc0 ls187 ws116">001031</div><div class="t m0 x5a ha y24d ffbb fs5 fc3 sc0 lsbe ws117">000152</div><div class="t m0 x5a ha y24e ffbb fs5 fc3 sc0 ls186">000000</div><div class="t m0 x26 h9 y246 ffba fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x26 h9 y247 ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x26 h9 y248 ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x26 h9 y249 ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x26 h9 y24a ffba fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa4 h6c y245 ffb9 fs5 fc3 sc0 lsb ws111">(d) <span class="ffba v1f">\u23a1</span></div><div class="t m0 x25 h9 y248 ffba fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x25 h9 y24f ffba fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x3d ha y250 ffbb fs5 fc3 sc0 ls162">1000</div><div class="t m0 x3d ha y251 ffbb fs5 fc3 sc0 ls162">0120</div><div class="t m0 x3d ha y252 ffbb fs5 fc3 sc0 ls162">0001</div><div class="t m0 x63 h9 y253 ffba fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x63 h9 y248 ffba fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x63 h9 y24f ffba fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x3f he y254 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 3<span class="_10 blank"> </span><span class="fs1 fc2 ls7a ws7e">Soluções de Quatro Sistemas</span></div><div class="t m0 x40 h3 y255 ff1 fs1 fc2 sc0 ls7a">Lineares</div><div class="t m0 x8f h9 y256 ffbd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x8f h9 y257 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y258 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y259 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y25a ffbd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xa5 ha y25b ffbe fs5 fc3 sc0 ls191">1000</div><div class="t m0 xa5 ha y25c ffbe fs5 fc3 sc0 ls18f">0100</div><div class="t m0 xa5 ha y25d ffbe fs5 fc3 sc0 ls191">0010</div><div class="t m0 xa5 ha y25e ffbe fs5 fc3 sc0 ls190">0001</div><div class="t m0 xa6 h9 y256 ffbd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa6 h9 y257 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y258 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y259 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y25a ffbd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x35 h6f y25f ffbf fs5 fc3 sc0 lsb">,</div><div class="t m0 xa7 h9 y256 ffbd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa7 h9 y257 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y258 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y259 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y25a ffbd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x45 ha y25b ffbe fs5 fc3 sc0 ls188 ws118">100<span class="ffc0 lsb v9">*</span></div><div class="t m0 x45 ha y25c ffbe fs5 fc3 sc0 ls189 ws119">010<span class="ffc0 lsb v9">*</span></div><div class="t m0 x45 ha y25d ffbe fs5 fc3 sc0 ls18a ws11a">001<span class="ffc0 lsb v9">*</span></div><div class="t m0 x45 ha y25e ffbe fs5 fc3 sc0 ls162">0000</div><div class="t m0 x5 h9 y256 ffbd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x5 h9 y257 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y258 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y259 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y25a ffbd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa8 h6f y25f ffbf fs5 fc3 sc0 lsb">,</div><div class="t m0 x8f h9 y260 ffbd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x8f h9 y261 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y262 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y263 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y264 ffbd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xa5 ha y265 ffbe fs5 fc3 sc0 ls19d ws11b">10<span class="ffc0 ls162 v9">**</span></div><div class="t m0 xa5 ha y266 ffbe fs5 fc3 sc0 ls19e ws11c">01<span class="ffc0 ls162 v9">**</span></div><div class="t m0 xa5 ha y267 ffbe fs5 fc3 sc0 ls162">0000</div><div class="t m0 xa5 ha y268 ffbe fs5 fc3 sc0 ls162">0000</div><div class="t m0 xa6 h9 y260 ffbd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa6 h9 y261 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y262 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y263 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y264 ffbd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x35 h6f y269 ffbf fs5 fc3 sc0 lsb">,</div><div class="t m0 xa7 h9 y26a ffbd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa7 h9 y26b ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y26c ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y26d ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y26e ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y26f ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y270 ffbd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y271 ffbd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x45 ha y272 ffbe fs5 fc3 sc0 ls19e ws11c">01<span class="ffc0 ls18b v9">*</span><span class="ls162 ws11d">000<span class="ffc0 v9">**</span><span class="ls18c">0<span class="ffc0 lsb v9">*</span></span></span></div><div class="t m0 x45 ha y273 ffbe fs5 fc3 sc0 ls162 ws11d">000100<span class="ffc0 v9">**</span><span class="ls18c">0<span class="ffc0 lsb v9">*</span></span></div><div class="t m0 x45 ha y274 ffbe fs5 fc3 sc0 ls18d ws11e">000010<span class="ffc0 ls162 ws11d v9">**</span><span class="ls18c">0<span class="ffc0 lsb v9">*</span></span></div><div class="t m0 x45 ha y275 ffbe fs5 fc3 sc0 ls18e">000001<span class="ffc0 ls162 ws11d v9">**</span><span class="ls18c">0<span class="ffc0 lsb v9">*</span></span></div><div class="t m0 x45 ha y276 ffbe fs5 fc3 sc0 ls18f ws11f">000000001<span class="ffc0 lsb v9">*</span></div><div class="t m0 xa9 h9 y26a ffbd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa9 h9 y26b ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y26c ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y26d ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y26e ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y26f ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y270 ffbd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y271 ffbd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x8f h9 y277 ffc1 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x8f h9 y278 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y279 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y27a ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y27b ffc1 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xa5 ha y27c ffc2 fs5 fc3 sc0 ls18c">1<span class="ffc3 ls188 v9">***</span></div><div class="t m0 xa5 ha y27d ffc2 fs5 fc3 sc0 ls19e ws11c">01<span class="ffc3 ls19d v9">**</span></div><div class="t m0 xa5 ha y27e ffc2 fs5 fc3 sc0 ls18a ws11a">001<span class="ffc3 lsb v9">*</span></div><div class="t m0 xa5 ha y27f ffc2 fs5 fc3 sc0 ls162">0001</div><div class="t m0 xa6 h9 y277 ffc1 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa6 h9 y278 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y279 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y27a ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y27b ffc1 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x35 h6f y280 ffc4 fs5 fc3 sc0 lsb">,</div><div class="t m0 xa7 h9 y277 ffc1 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa7 h9 y278 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y279 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y27a ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y27b ffc1 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x45 ha y27c ffc2 fs5 fc3 sc0 ls18c">1<span class="ffc3 ls162 v9">***</span></div><div class="t m0 x45 ha y27d ffc2 fs5 fc3 sc0 ls19e ws11c">01<span class="ffc3 ls162 v9">**</span></div><div class="t m0 x45 ha y27e ffc2 fs5 fc3 sc0 ls18a ws11a">001<span class="ffc3 lsb v9">*</span></div><div class="t m0 x45 ha y27f ffc2 fs5 fc3 sc0 ls162">0000</div><div class="t m0 x5 h9 y277 ffc1 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x5 h9 y278 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y279 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y27a ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5 h9 y27b ffc1 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa8 h6f y280 ffc4 fs5 fc3 sc0 lsb">,</div><div class="t m0 x8f h9 y281 ffc1 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x8f h9 y282 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y283 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y284 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8f h9 y285 ffc1 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xa5 ha y286 ffc2 fs5 fc3 sc0 ls18c">1<span class="ffc3 ls162 v9">***</span></div><div class="t m0 xa5 ha y287 ffc2 fs5 fc3 sc0 ls19e ws11c">01<span class="ffc3 ls162 v9">**</span></div><div class="t m0 xa5 ha y288 ffc2 fs5 fc3 sc0 ls162">0000</div><div class="t m0 xa5 ha y289 ffc2 fs5 fc3 sc0 ls162">0000</div><div class="t m0 xa6 h9 y281 ffc1 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa6 h9 y282 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y283 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y284 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa6 h9 y285 ffc1 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x35 h6f y28a ffc4 fs5 fc3 sc0 lsb">,</div><div class="t m0 xa7 h9 y28b ffc1 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa7 h9 y28c ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y28d ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y28e ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y28f ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y290 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y291 ffc1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa7 h9 y292 ffc1 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x45 ha y293 ffc2 fs5 fc3 sc0 ls19e ws11c">01<span class="ffc3 ls18d ws11e v9">********</span></div><div class="t m0 x45 ha y294 ffc2 fs5 fc3 sc0 ls190 ws120">0001<span class="ffc3 ls191 ws121 v9">******</span></div><div class="t m0 x45 ha y295 ffc2 fs5 fc3 sc0 ls192 ws122">00001<span class="ffc3 ls19f v9">*****</span></div><div class="t m0 x45 ha y296 ffc2 fs5 fc3 sc0 ls18e">000001<span class="ffc3 ls19d v9">****</span></div><div class="t m0 x45 ha y297 ffc2 fs5 fc3 sc0 ls162 ws11d">000000001<span class="ffc3 lsb v9">*</span></div><div class="t m0 xa9 h9 y28b ffc1 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xa9 h9 y28c ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y28d ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y28e ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y28f ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y290 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y291 ffc1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xa9 h9 y292 ffc1 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x2 he y298 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 2<span class="_10 blank"> </span><span class="fs1 fc2 ls7a ws7e">Mais sobre Formas Escalonada e</span></div><div class="t m0 xaa h3 y299 ff1 fs1 fc2 sc0 ls7a ws7e">Escalonada Reduzida por Linhas</div><div class="t m0 xab h9 y29a ffc5 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xab h9 y29b ffc5 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x13 ha y29c ffc6 fs5 fc3 sc0 lsbd">1004</div><div class="t m0 x13 ha y29d ffc6 fs5 fc3 sc0 ls130 ws113">0107</div><div class="t m0 x13 ha y29e ffc6 fs5 fc3 sc0 lsc2 wsd4">001<span class="_15 blank"></span><span class="ffc7 lsb">\u2212<span class="ffc6">1</span></span></div><div class="t m0 x38 h9 y29a ffc5 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x38 h70 y29b ffc5 fs5 fc3 sc0 lsb ws24">\u23a6<span class="ffc8 ls193 v13">,</span><span class="v20">\u23a1</span></div><div class="t m0 x45 h9 y29f ffc5 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x45 h9 y2a0 ffc5 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xac ha y2a1 ffc6 fs5 fc3 sc0 lsdb ws123">100</div><div class="t m0 xac ha y2a2 ffc6 fs5 fc3 sc0 ls194 ws124">010</div><div class="t m0 xac ha y2a3 ffc6 fs5 fc3 sc0 ls18c ws125">001</div><div class="t m0 xad h9 y2a4 ffc5 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xad h9 y29f ffc5 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xad h71 y2a0 ffc5 fs5 fc3 sc0 lsb ws24">\u23a6<span class="ffc8 ls195 v17">,</span><span class="v18">\u23a1</span></div><div class="t m0 x8a h9 y2a5 ffc5 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8a h9 y2a6 ffc5 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8a h9 y2a7 ffc5 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x78 ha y2a8 ffc6 fs5 fc3 sc0 lsc2 wsd4">01<span class="_15 blank"></span><span class="ffc7 lsb">\u2212<span class="ffc6 ls1a0">201</span></span></div><div class="t m0 x78 ha y2a9 ffc6 fs5 fc3 sc0 ls196 ws126">00013</div><div class="t m0 x78 ha y2aa ffc6 fs5 fc3 sc0 ls1a1">00000</div><div class="t m0 x78 ha y2ab ffc6 fs5 fc3 sc0 ls1a1">00000</div><div class="t m0 xae h9 y2ac ffc5 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xae h9 y2a5 ffc5 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xae h9 y2a6 ffc5 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xae h9 y2a7 ffc5 fs5 fc3 sc0 lsb">\u23a6</div><div class="c xab y2ad wc h72"><div class="t m0 x8c h73 y2ae ffc8 fs5 fc3 sc0 lsb">,</div></div><div class="t m0 x1 h74 y2af ffc9 fs1a fc3 sc0 lsb">\u23a1</div><div class="t m0 x1 h74 y2b0 ffc9 fs1a fc3 sc0 lsb">\u23a3</div><div class="t m0 x7a h75 y2b1 ffca fs1a fc3 sc0 ls1a2">1004</div><div class="t m0 x7a h75 y2b2 ffca fs1a fc3 sc0 ls1a3 ws127">0107</div><div class="t m0 x7a h75 y2b3 ffca fs1a fc3 sc0 ls1a4 ws128">001<span class="_2c blank"></span><span class="ffcb lsb">\u2212<span class="ffca">1</span></span></div><div class="t m0 xaf h74 y2af ffc9 fs1a fc3 sc0 lsb">\u23a4</div><div class="t m0 xaf h76 y2b0 ffc9 fs1a fc3 sc0 lsb ws129">\u23a6<span class="ffcc ls197 v13">,</span><span class="v19">\u23a1</span></div><div class="t m0 x6a h74 y2b4 ffc9 fs1a fc3 sc0 lsb">\u23a2</div><div class="t m0 x6a h74 y2b5 ffc9 fs1a fc3 sc0 lsb">\u23a3</div><div class="t m0 x34 h75 y2b6 ffca fs1a fc3 sc0 ls198 ws12a">100</div><div class="t m0 x34 h75 y2b7 ffca fs1a fc3 sc0 ls199 ws12b">010</div><div class="t m0 x34 h75 y2b8 ffca fs1a fc3 sc0 ls19a ws12c">001</div><div class="t m0 x82 h74 y2b9 ffc9 fs1a fc3 sc0 lsb">\u23a4</div><div class="t m0 x82 h74 y2b4 ffc9 fs1a fc3 sc0 lsb">\u23a5</div><div class="t m0 x82 h77 y2b5 ffc9 fs1a fc3 sc0 lsb ws129">\u23a6<span class="ffcc ls197 v17">,</span><span class="v21">\u23a1</span></div><div class="t m0 xad h74 y2ba ffc9 fs1a fc3 sc0 lsb">\u23a2</div><div class="t m0 xad h74 y2bb ffc9 fs1a fc3 sc0 lsb">\u23a2</div><div class="t m0 xad h74 y2bc ffc9 fs1a fc3 sc0 lsb">\u23a3</div><div class="t m0 xb0 h75 y2bd ffca fs1a fc3 sc0 ls1a4 ws128">01<span class="_2c blank"></span><span class="ffcb lsb">\u2212<span class="ffca ls1a5">201</span></span></div><div class="t m0 xb0 h75 y2be ffca fs1a fc3 sc0 ls19b ws12d">00013</div><div class="t m0 xb0 h75 y2bf ffca fs1a fc3 sc0 ls1a6">00000</div><div class="t m0 xb0 h75 y2c0 ffca fs1a fc3 sc0 ls1a6">00000</div><div class="t m0 xb1 h74 y2c1 ffc9 fs1a fc3 sc0 lsb">\u23a4</div><div class="t m0 xb1 h74 y2ba ffc9 fs1a fc3 sc0 lsb">\u23a5</div><div class="t m0 xb1 h74 y2bb ffc9 fs1a fc3 sc0 lsb">\u23a5</div><div class="t m0 xb1 h74 y2bc ffc9 fs1a fc3 sc0 lsb">\u23a6</div><div class="t m0 xb2 h78 y2c2 ffcc fs1a fc3 sc0 ls197">,<span class="ffc9 lsb ws129 v22">\ue008</span><span class="ffca ls1a7 v1d">00</span></div><div class="t m0 xb3 h75 y2c3 ffca fs1a fc3 sc0 ls1a7">00</div><div class="t m0 x91 h74 y2c4 ffc9 fs1a fc3 sc0 lsb">\ue009</div><div class="t m0 x2 he y2c5 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 1<span class="_10 blank"> </span><span class="fs1 fc2 ls7a ws7e">Formas Escalonada e Escalonada</span></div><div class="t m0 xaa h3 y2c6 ff1 fs1 fc2 sc0 ls7a ws7e">Reduzida por Linhas</div><div class="t m0 xb h2a ya3 ff1 fs7 fc1 sc0 lsd ws2e">32 <span class="fsf ls7c ws7f vf">\u2022 \u2022 \u2022<span class="_12 blank"> </span></span><span class="ff3 fs10 fc2 ls7d ws80">Álgebra Linear com <span class="_0 blank"></span>Aplicações</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y2c7 w9 h79" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg7.png"><div class="t m0 xb h4 ya5 ff3 fs2 fc2 sc0 lsb ws160">Como a <span class="ff4 ws17">x</span><span class="fs3 ls1a8 v6">5</span><span class="lsa ws161">pode ser atribuído um valor arbitrário </span><span class="ff4">t</span>, e a <span class="ff4 ws17">x</span><span class="fs3 ws7 v6">2 <span class="_4 blank"> </span></span>pode</div><div class="t m0 xb h4 y2c8 ff3 fs2 fc2 sc0 lsa ws162">ser atribuído um valor arbitrário<span class="_13 blank"> </span><span class="ff4 lsb">s</span>, existem infinitas soluções. <span class="_5 blank"></span>A</div><div class="t m0 xb h4 y2c9 ff3 fs2 fc2 sc0 lsa ws6">solução geral é dada pelas fórmulas</div><div class="t m0 xb h4 y2ca ff4 fs2 fc1 sc0 lsb ws6">Solução <span class="ff3">(</span>d<span class="ff3 ls72 ws148">). <span class="fc2 lsb ws3c">A<span class="_25 blank"> </span>última equação do sistema correspondente é</span></span></div><div class="t m0 xb h4 y2cb ff3 fs2 fc2 sc0 lsb ws163">Como esta equação não pode ser resolvida, não existe solução</div><div class="t m0 xb h14 y2cc ff3 fs2 fc2 sc0 lsa ws6">para o sistema.<span class="_2d blank"> </span><span class="ffe fc1 lsb">®</span></div><div class="t m0 xb he y2cd ff1 fs7 fc1 sc0 ls71 ws164">Métodos de Eliminação <span class="ff3 fs2 fc2 lsa ws165">Nós acabamos de ver como é</span></div><div class="t m0 xb h4 y2ce ff3 fs2 fc2 sc0 lsa ws166">fácil resolver um sistema de equações lineares tão logo sua</div><div class="t m0 xb h4 y2cf ff3 fs2 fc2 sc0 lsb ws167">matriz aumentada estiver em forma escalonada reduzida por li-</div><div class="t m0 xb h4 y2d0 ff3 fs2 fc2 sc0 lsa ws168">nhas. <span class="_0 blank"></span>Agora nós iremos dar um procedimento de <span class="ffd lsb">eliminação</span></div><div class="t m0 xb h4 y2d1 ff3 fs2 fc2 sc0 lsa ws169">passo a passo que pode ser usado para reduzir qualquer matriz à</div><div class="t m0 xb h4 y2d2 ff3 fs2 fc2 sc0 lsb ws16a">forma escalonada. À medida que enunciamos cada passo, ire-</div><div class="t m0 xb h4 y2d3 ff3 fs2 fc2 sc0 lsa ws16b">mos ilustrá-lo reduzindo a seguinte matriz à forma escalonada</div><div class="t m0 xb h4 y2d4 ff3 fs2 fc2 sc0 lsa ws6">reduzida por linhas.</div><div class="t m0 xb h2d y2d5 ff1 fs2 fc2 sc0 lsc ws4e">Passo 1.<span class="_14 blank"> </span><span class="ff3 lsb ws16c">Localize a coluna mais à esquerda que não seja cons-</span></div><div class="t m0 xab h4 y2d6 ff3 fs2 fc2 sc0 lsb ws3c">tituída inteiramente de zeros.</div><div class="t m0 xb4 h7a y2d7 ff2 fsa fc2 sc0 ls73 ws76">Coluna não-nula mais à esquerda</div><div class="t m0 xb h2d y2d8 ff1 fs2 fc2 sc0 lsc ws16d">Passo 2.<span class="_2e blank"> </span><span class="ff3 lsb ws16e">Permute a primeira linha com uma outra linha, se</span></div><div class="t m0 xab h4 y2d9 ff3 fs2 fc2 sc0 lsb ws16f">necessário, para obter uma entrada não-nula ao topo da colu-</div><div class="t m0 xab h4 y2da ff3 fs2 fc2 sc0 lsa ws6">na encontrada no Passo 1.</div><div class="t m0 xb h2d y2db ff1 fs2 fc2 sc0 lsc ws170">Passo 3.<span class="_14 blank"> </span><span class="ff3 lsb ws171">Se a entrada que agora está no topo da coluna encon-</span></div><div class="t m0 xab h4 y2dc ff3 fs2 fc2 sc0 lsa ws172">trada no Passo 1 é <span class="ff4 lsb">a<span class="ff3 ws173">, multiplique a primeira linha inteira por</span></span></div><div class="t m0 xab h4 y2dd ff3 fs2 fc2 sc0 lsa">1/<span class="ff4 ls42">a</span><span class="ws6">para introduzir um líder<span class="_0 blank"></span>.</span></div><div class="t m0 xb h2d y2de ff1 fs2 fc2 sc0 lsc ws174">Passo 4.<span class="_2f blank"> </span><span class="ff3 lsb ws175">Some múltiplos convenientes da primeira linha às li-</span></div><div class="t m0 xab h4 y2df ff3 fs2 fc2 sc0 lsa ws176">nhas inferiores para obter zeros em todas as entradas abaixo</div><div class="t m0 xab h4 y2e0 ff3 fs2 fc2 sc0 lsa ws6">do líder<span class="_0 blank"></span>.</div><div class="t m0 xb h2d y2e1 ff1 fs2 fc2 sc0 lsc ws177">Passo 5. <span class="ff3 lsb ws178">Agora esconda a primeira linha da matriz e recomece</span></div><div class="t m0 xab h4 y2e2 ff3 fs2 fc2 sc0 lsb ws179">aplicando o Passo 1 à submatriz resultante. Continue desta</div><div class="t m0 xab h4 y2e3 ff3 fs2 fc2 sc0 lsb ws3c">maneira até que <span class="ff4 ws6">toda </span>a matriz esteja em forma escalonada.</div><div class="t m0 xb5 h7a y2e4 ff2 fsa fc2 sc0 ls73 ws76">Coluna não-nula mais à</div><div class="t m0 xb5 h7a y2e5 ff2 fsa fc2 sc0 ls73 ws76">esquerda da submatriz.</div><div class="t m0 xb6 h7a y2e6 ff2 fsa fc2 sc0 ls73 ws17a">Coluna não-nula mais à esquerda</div><div class="t m0 xb6 h7a y2e7 ff2 fsa fc2 sc0 ls73 ws9b">da nova submatriz.</div><div class="t m0 x14 h4 y2e8 ff3 fs2 fc2 sc0 lsa ws7">A<span class="_27 blank"> </span>matriz <span class="_13 blank"> </span><span class="ff4 lsb ws149">toda </span><span class="ws17b">está agora em forma escalonada. Para obter a</span></div><div class="t m0 x14 h4 y2e9 ff3 fs2 fc2 sc0 lsa ws17c">forma escalonada reduzida por linhas precisamos de mais um</div><div class="t m0 x14 h4 y2ea ff3 fs2 fc2 sc0 lsa">passo.</div><div class="t m0 x14 h2d y2eb ff1 fs2 fc2 sc0 lsc ws17d">Passo 6.<span class="_14 blank"> </span><span class="ff3 lsb ws171">Começando com a última linha não-nula e trabalhan-</span></div><div class="t m0 x85 h4 y2ec ff3 fs2 fc2 sc0 lsb ws17e">do para cima, some múltiplos convenientes de cada linha às</div><div class="t m0 x85 h4 y2ed ff3 fs2 fc2 sc0 lsa ws6">linhas superiores para introduzir zeros acima dos líderes.</div><div class="t m0 x14 h4 y2ee ff3 fs2 fc2 sc0 lsb ws6">A<span class="_25 blank"> </span>última matriz está na forma escalonada reduzida por linhas.</div><div class="t m0 x15 h4 y2ef ff3 fs2 fc2 sc0 lsa ws17f">Se nós usarmos somente os cinco primeiros passos, o pro-</div><div class="t m0 x14 h4 y2f0 ff3 fs2 fc2 sc0 lsb ws64">cedimento acima, chamado <span class="ffd lsa">eliminação gaussiana<span class="ff3 ws180">, produzirá</span></span></div><div class="t m0 x14 h4 y2f1 ff3 fs2 fc2 sc0 lsb ws181">uma forma escalonada. O procedimento até o sexto passo, que</div><div class="t m0 x14 h4 y2f2 ff3 fs2 fc2 sc0 lsb ws182">produz uma matriz em forma escalonada reduzida por linhas, é</div><div class="t m0 x14 h4 y2f3 ff3 fs2 fc2 sc0 lsb ws7">chamado <span class="ffd lsa ws6">eliminação de Gauss-Jordan</span>.</div><div class="t m0 x14 h4 y2f4 ff3 fs11 fc1 sc0 ls1b6 ws14a">OBSER<span class="_0 blank"></span>V<span class="_5 blank"></span>AÇÃO<span class="fs2 ls1a9">.<span class="fc2 lsa ws183">Pode ser mostrado que <span class="ff4 lsba ws184">toda matriz tem uma única</span></span></span></div><div class="t m0 x14 h4 y2f5 ff4 fs2 fc2 sc0 lsa ws185">forma escalonada r<span class="_0 blank"></span>eduzida por linhas<span class="ff3 lsb">; ou seja, sempre che-</span></div><div class="t m0 x14 h4 y2f6 ff3 fs2 fc2 sc0 lsa ws186">gamos à mesma forma escalonada reduzida por linhas para uma</div><div class="t m0 x14 h4 y2f7 ff3 fs2 fc2 sc0 lsb ws142">dada matriz, não importa como variamos as operações sobre li-</div><div class="t m0 x14 h4 y2f8 ff3 fs2 fc2 sc0 lsa ws187">nhas. (Uma prova deste resultado pode ser encontrada no artigo</div><div class="t m0 x14 h4 y2f9 ff3 fs2 fc2 sc0 lsa ws188">\u201cThe Reduced Row Echeleon Form of a Matrix Is Unique: <span class="_0 blank"></span>A</div><div class="t m0 x24 h7b y2fa ff3 fs1b fc1 sc0 ls1b7 ws189">5 vezes a segunda linha foi</div><div class="t m0 x24 h7b y2fb ff3 fs1b fc1 sc0 ls1b7 ws189">somada à primeira linha</div><div class="t m0 x3 h9 y2fc ffcd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x3 h9 y2fd ffcd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3 h9 y2fe ffcd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xb7 ha y2ff ffce fs5 fc3 sc0 lsc2 wsd4">120307</div><div class="t m0 xb7 ha y300 ffce fs5 fc3 sc0 ls1aa ws14b">001001</div><div class="t m0 xb7 ha y301 ffce fs5 fc3 sc0 ls1ab ws14c">000012</div><div class="t m0 x1f h9 y2fc ffcd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x1f h9 y2fd ffcd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x1f h9 y2fe ffcd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x24 h7b y302 ff3 fs1b fc1 sc0 ls1b7 ws189">\u20136 vezes a terceira linha foi</div><div class="t m0 x24 h7b y303 ff3 fs1b fc1 sc0 lsb ws189">somada à primeira linha.</div><div class="t m0 x3 h9 y304 ffcf fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x3 h9 y305 ffcf fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3 h9 y306 ffcf fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xb7 ha y307 ffd0 fs5 fc3 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffd1 lsb">\u2212<span class="ffd0 ls134 ws14d">5302</span></span></div><div class="t m0 xb7 ha y308 ffd0 fs5 fc3 sc0 ls1ac ws14e">001001</div><div class="t m0 xb7 ha y309 ffd0 fs5 fc3 sc0 ls1ad ws14f">000012</div><div class="t m0 x28 h9 y304 ffcf fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x28 h9 y305 ffcf fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x28 h9 y306 ffcf fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x24 h7b y30a ff3 fs1b fc1 sc0 lsb ws18a">7/2 vezes a terceira linha da</div><div class="t m0 x24 h7b y30b ff3 fs1b fc1 sc0 ls1b7 ws18a">matriz precedente foi somada à</div><div class="t m0 x24 h7b y30c ff3 fs1b fc1 sc0 lsb ws18a">segunda linha.</div><div class="t m0 x3 h9 y30d ffd2 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x3 h9 y30e ffd2 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x3 h9 y30f ffd2 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xb7 ha y310 ffd3 fs5 fc3 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffd4 lsb">\u2212<span class="ffd3 lsbe">5361<span class="_8 blank"></span>4</span></span></div><div class="t m0 xb7 ha y311 ffd3 fs5 fc3 sc0 ls1a0 ws150">00100 1</div><div class="t m0 xb7 ha y312 ffd3 fs5 fc3 sc0 ls1ae ws151">00001 2</div><div class="t m0 x98 h9 y30d ffd2 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x98 h9 y30e ffd2 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x98 h9 y30f ffd2 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xb8 h7b y313 ff3 fs1b fc1 sc0 ls1b7 ws189">A<span class="_6 blank"> </span>primeira (e única) linha da</div><div class="t m0 xb8 h7b y314 ff3 fs1b fc1 sc0 lsb ws189">nova submatriz foi multiplicada</div><div class="t m0 xb8 h7b y315 ff3 fs1b fc1 sc0 ls1b7 ws189">por 2 para introduzir um líder<span class="_0 blank"></span>.</div><div class="t m0 xb9 h9 y316 ffd5 fs5 fc2 sc0 lsb">\u23a1</div><div class="t m0 xb9 h9 y317 ffd5 fs5 fc2 sc0 lsb">\u23a2</div><div class="t m0 xb9 h9 y318 ffd5 fs5 fc2 sc0 lsb">\u23a3</div><div class="t m0 xba ha y319 ffd6 fs5 fc2 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffd7 lsb">\u2212<span class="ffd6 lsbe ws152">53 6<span class="_30 blank"></span>1<span class="_8 blank"></span>4</span></span></div><div class="t m0 xba h7c y31a ffd6 fs5 fc2 sc0 lsbe ws86">0010<span class="_15 blank"></span><span class="ffd7 ls1af">\u2212<span class="ffd6 fs1c lsb v5">7</span></span></div><div class="t m0 x9f h7d y31b ffd6 fs1c fc2 sc0 lsb">2</div><div class="t m0 xbb ha y31c ffd7 fs5 fc2 sc0 lsb">\u2212<span class="ffd6">6</span></div><div class="t m0 xba ha y31d ffd6 fs5 fc2 sc0 lsbe ws152">0000 12</div><div class="t m0 x1e h9 y31e ffd5 fs5 fc2 sc0 lsb">\u23a4</div><div class="t m0 x1e h9 y31f ffd5 fs5 fc2 sc0 lsb">\u23a5</div><div class="t m0 x1e h9 y320 ffd5 fs5 fc2 sc0 lsb">\u23a6</div><div class="t m0 xb8 h7b y321 ff3 fs1b fc1 sc0 ls1b7 ws189">A<span class="_6 blank"> </span>linha superior da submatriz</div><div class="t m0 xb8 h7b y322 ff3 fs1b fc1 sc0 ls1b7 ws189">foi tratada e retornamos ao</div><div class="t m0 xb8 h7b y323 ff3 fs1b fc1 sc0 ls1b7 ws189">Passo 1.</div><div class="t m0 xb9 h9 y324 ffd8 fs5 fc2 sc0 lsb">\u23a1</div><div class="t m0 xb9 h9 y325 ffd8 fs5 fc2 sc0 lsb">\u23a2</div><div class="t m0 xb9 h9 y326 ffd8 fs5 fc2 sc0 lsb">\u23a3</div><div class="t m0 xba ha y327 ffd9 fs5 fc2 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffda lsb">\u2212<span class="ffd9 lsbe ws152">53 6<span class="_30 blank"></span>1<span class="_8 blank"></span>4</span></span></div><div class="t m0 xba h7e y328 ffd9 fs5 fc2 sc0 lsbe ws86">0010<span class="_15 blank"></span><span class="ffda ls1af">\u2212<span class="ffd9 fs1c lsb v5">7</span></span></div><div class="t m0 xa2 h7d y329 ffd9 fs1c fc2 sc0 lsb">2</div><div class="t m0 xbb ha y32a ffda fs5 fc2 sc0 lsb">\u2212<span class="ffd9">6</span></div><div class="t m0 xba h7f y32b ffd9 fs5 fc2 sc0 lsbe ws153">0000 <span class="fs1c lsb v5">1</span></div><div class="t m0 xa2 h80 y32c ffd9 fs1c fc2 sc0 ls1b0">2<span class="fs5 lsb v5">1</span></div><div class="t m0 x1e h9 y32d ffd8 fs5 fc2 sc0 lsb">\u23a4</div><div class="t m0 x1e h9 y32e ffd8 fs5 fc2 sc0 lsb">\u23a5</div><div class="t m0 x1e h9 y32f ffd8 fs5 fc2 sc0 lsb">\u23a6</div><div class="t m0 xb8 h7b y330 ff3 fs1b fc1 sc0 lsb ws18a">\u20135 vezes a primeira linha da</div><div class="t m0 xb8 h7b y331 ff3 fs1b fc1 sc0 ls1b7 ws18a">submatriz foi somada à segunda</div><div class="t m0 xb8 h7b y332 ff3 fs1b fc1 sc0 ls1b7 ws18a">linha da submatriz para intro-</div><div class="t m0 xb8 h7b y333 ff3 fs1b fc1 sc0 lsb ws18a">duzir um zero debaixo do líder<span class="_0 blank"></span>.</div><div class="t m0 xb9 h9 y334 ffdb fs5 fc2 sc0 lsb">\u23a1</div><div class="t m0 xb9 h9 y335 ffdb fs5 fc2 sc0 lsb">\u23a2</div><div class="t m0 xb9 h9 y336 ffdb fs5 fc2 sc0 lsb">\u23a3</div><div class="t m0 xba ha y337 ffdc fs5 fc2 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffdd lsb">\u2212<span class="ffdc lsbe ws152">53 6<span class="_30 blank"></span>1<span class="_8 blank"></span>4</span></span></div><div class="t m0 xba h81 y338 ffdc fs5 fc2 sc0 lsbe ws86">0010<span class="_15 blank"></span><span class="ffdd ls1af">\u2212<span class="ffdc fs1c lsb v5">7</span></span></div><div class="t m0 xa2 h7d y339 ffdc fs1c fc2 sc0 lsb">2</div><div class="t m0 xbb ha y33a ffdd fs5 fc2 sc0 lsb">\u2212<span class="ffdc">6</span></div><div class="t m0 xba h82 y33b ffdc fs5 fc2 sc0 lsbe ws153">0000 <span class="fs1c lsb v5">1</span></div><div class="t m0 xa2 h80 y33c ffdc fs1c fc2 sc0 ls1b0">2<span class="fs5 lsb v5">1</span></div><div class="t m0 x1e h9 y33d ffdb fs5 fc2 sc0 lsb">\u23a4</div><div class="t m0 x1e h9 y33e ffdb fs5 fc2 sc0 lsb">\u23a5</div><div class="t m0 x1e h9 y33f ffdb fs5 fc2 sc0 lsb">\u23a6</div><div class="t m0 xb8 h7b y340 ff3 fs1b fc1 sc0 ls1b7 ws189">A<span class="_6 blank"> </span>primeira linha da submatriz foi</div><div class="t m0 xb8 h7b y341 ff3 fs1b fc1 sc0 lsb ws189">multiplicada por \u20131/2 para intro-</div><div class="t m0 xb8 h7b y342 ff3 fs1b fc1 sc0 ls1b7 ws189">duzir um líder<span class="_0 blank"></span>.</div><div class="t m0 x3f h9 y343 ffde fs5 fc2 sc0 lsb">\u23a1</div><div class="t m0 x3f h9 y344 ffde fs5 fc2 sc0 lsb">\u23a2</div><div class="t m0 x3f h9 y345 ffde fs5 fc2 sc0 lsb">\u23a3</div><div class="t m0 x3 ha y346 ffdf fs5 fc2 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffe0 lsb">\u2212<span class="ffdf lsbe ws154">53 61<span class="_8 blank"></span>4</span></span></div><div class="t m0 x3 h83 y347 ffdf fs5 fc2 sc0 lsbe ws86">0010<span class="_28 blank"></span><span class="ffe0 ls1b1">\u2212<span class="ffdf fs1c lsb v5">7</span></span></div><div class="t m0 xbc h7d y348 ffdf fs1c fc2 sc0 lsb">2</div><div class="t m0 xbb ha y349 ffe0 fs5 fc2 sc0 lsb">\u2212<span class="ffdf">6</span></div><div class="t m0 x3 ha y34a ffdf fs5 fc2 sc0 lsbe ws86">0050<span class="_15 blank"></span><span class="ffe0 lsb">\u2212<span class="ffdf wsd5">17 </span>\u2212<span class="ffdf">29</span></span></div><div class="t m0 x1e h9 y34b ffde fs5 fc2 sc0 lsb">\u23a4</div><div class="t m0 x1e h9 y34c ffde fs5 fc2 sc0 lsb">\u23a5</div><div class="t m0 x1e h9 y34d ffde fs5 fc2 sc0 lsb">\u23a6</div><div class="c x3f y34e wd h84"><div class="t m0 x46 h85 y34f ffe1 fsc fc1 sc0 lsb">m</div></div><div class="t m0 xbd h9 y350 ffe2 fs5 fc2 sc0 lsb">\u23a1</div><div class="t m0 xbd h9 y351 ffe2 fs5 fc2 sc0 lsb">\u23a2</div><div class="t m0 xbd h9 y352 ffe2 fs5 fc2 sc0 lsb">\u23a3</div><div class="t m0 x97 ha y353 ffe3 fs5 fc2 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffe4 lsb">\u2212<span class="ffe3 lsbe ws154">53 61<span class="_8 blank"></span>4</span></span></div><div class="t m0 x97 ha y354 ffe3 fs5 fc2 sc0 lsbe ws86">00<span class="_15 blank"></span><span class="ffe4 lsb">\u2212<span class="ffe3 lsbe ws154">20 71<span class="_8 blank"></span>2</span></span></div><div class="t m0 x97 ha y355 ffe3 fs5 fc2 sc0 lsbe ws86">0050<span class="_15 blank"></span><span class="ffe4 lsb">\u2212<span class="ffe3 wsd5">17 </span>\u2212<span class="ffe3">29</span></span></div><div class="t m0 xbe h9 y350 ffe2 fs5 fc2 sc0 lsb">\u23a4</div><div class="t m0 xbe h9 y351 ffe2 fs5 fc2 sc0 lsb">\u23a5</div><div class="t m0 xbe h9 y352 ffe2 fs5 fc2 sc0 lsb">\u23a6</div><div class="t m0 x74 h7b y356 ff3 fs1b fc1 sc0 lsb ws18a">\u20132 vezes a primeira linha da</div><div class="t m0 x74 h7b y357 ff3 fs1b fc1 sc0 lsb ws18a">matriz precedente foi somada à</div><div class="t m0 x74 h7b y358 ff3 fs1b fc1 sc0 ls1b7 ws18a">terceira linha.</div><div class="t m0 x8d h9 y359 ffe5 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x8d h9 y35a ffe5 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x8d h9 y35b ffe5 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xab ha y35c ffe6 fs5 fc3 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffe7 lsb">\u2212<span class="ffe6 ls1b2 ws155">53 61<span class="_8 blank"></span>4</span></span></div><div class="t m0 xab ha y35d ffe6 fs5 fc3 sc0 ls1b8 ws156">00<span class="_15 blank"></span><span class="ffe7 lsb">\u2212<span class="ffe6 ls1b3 ws157">20 71<span class="_8 blank"></span>2</span></span></div><div class="t m0 xab ha y35e ffe6 fs5 fc3 sc0 ls1b8 ws156">0050<span class="_15 blank"></span><span class="ffe7 lsb">\u2212<span class="ffe6 wsd5">17 </span>\u2212<span class="ffe6">29</span></span></div><div class="t m0 x82 h9 y359 ffe5 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x82 h9 y35a ffe5 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x82 h9 y35b ffe5 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x48 h7b y35f ff3 fs1b fc1 sc0 lsb ws18a">A<span class="_6 blank"> </span>primeira linha da matriz</div><div class="t m0 x48 h7b y360 ff3 fs1b fc1 sc0 lsb ws18a">precedente foi multiplicada por</div><div class="t m0 x48 h7b y361 ff3 fs1b fc1 sc0 lsb">1/2.</div><div class="t m0 x4d h9 y362 ffe8 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x4d h9 y363 ffe8 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x4d h9 y364 ffe8 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x79 ha y365 ffe9 fs5 fc3 sc0 lsbe ws86">12<span class="_15 blank"></span><span class="ffea lsb">\u2212<span class="ffe9 ls134 ws14d">536<span class="_30 blank"></span>1<span class="_8 blank"></span>4</span></span></div><div class="t m0 xbf ha y366 ffe9 fs5 fc3 sc0 ls1b8 ws156">00<span class="_15 blank"></span><span class="ffea lsb">\u2212<span class="ffe9 ls1b2 ws158">207<span class="_30 blank"></span>1<span class="_8 blank"></span>2</span></span></div><div class="t m0 xbf ha y367 ffe9 fs5 fc3 sc0 ls1b9 ws159">24<span class="_15 blank"></span><span class="ffea lsb">\u2212<span class="ffe9 lsbe ws86">56<span class="_15 blank"></span><span class="ffea lsb">\u2212<span class="ffe9 lsdb">5</span>\u2212<span class="ffe9">1</span></span></span></span></div><div class="t m0 xc0 h9 y362 ffe8 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc0 h9 y363 ffe8 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc0 h9 y364 ffe8 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x31 h7b y368 ff3 fs1b fc1 sc0 ls1b7 ws189">Foram permutadas a primeira e</div><div class="t m0 x31 h7b y369 ff3 fs1b fc1 sc0 lsb ws189">a segunda linhas da matriz</div><div class="t m0 x31 h7b y36a ff3 fs1b fc1 sc0 lsb">precedente.</div><div class="t m0 x39 h9 y36b ffeb fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x39 h9 y36c ffeb fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x39 h9 y36d ffeb fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xc1 ha y36e ffec fs5 fc3 sc0 ls1b8 ws156">24<span class="_15 blank"></span><span class="ffed lsb">\u2212<span class="ffec ws15a">10<span class="_31 blank"> </span>6 12 28</span></span></div><div class="t m0 xc1 ha y36f ffec fs5 fc3 sc0 ls1ba ws15b">00<span class="_d blank"></span><span class="ffed lsb">\u2212<span class="ffec ls1b2 ws158">207<span class="_30 blank"></span>1<span class="_8 blank"></span>2</span></span></div><div class="t m0 xc1 ha y370 ffec fs5 fc3 sc0 ls1b8 ws156">24<span class="_d blank"></span><span class="ffed lsb">\u2212<span class="ffec lsbe ws86">56<span class="_15 blank"></span><span class="ffed lsb">\u2212<span class="ffec lsdb">5</span>\u2212<span class="ffec">1</span></span></span></span></div><div class="t m0 xc2 h9 y36b ffeb fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc2 h9 y36c ffeb fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc2 h9 y36d ffeb fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa6 h9 y371 ffee fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa6 h9 y372 ffee fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa6 h9 y373 ffee fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x35 ha y374 ffef fs5 fc3 sc0 lsbe ws86">00<span class="_d blank"></span><span class="fff0 lsb">\u2212<span class="ffef lsbe">207<span class="_30 blank"></span>1<span class="_8 blank"></span>2</span></span></div><div class="t m0 x35 ha y375 ffef fs5 fc3 sc0 lsbe ws86">24<span class="_15 blank"></span><span class="fff0 lsb">\u2212<span class="ffef ws15c">10<span class="_31 blank"> </span>6 12 28</span></span></div><div class="t m0 x35 ha y376 ffef fs5 fc3 sc0 lsbe ws86">24<span class="_d blank"></span><span class="fff0 lsb">\u2212<span class="ffef lsbe">56<span class="_15 blank"></span><span class="fff0 lsb">\u2212<span class="ffef lsdb">5</span>\u2212<span class="ffef">1</span></span></span></span></div><div class="t m0 xc3 h9 y371 ffee fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc3 h9 y372 ffee fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc3 h9 y373 ffee fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa6 h9 y377 fff1 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xa6 h9 y378 fff1 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xa6 h9 y379 fff1 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x35 ha y37a fff2 fs5 fc3 sc0 lsc2 wsd4">00<span class="_d blank"></span><span class="fff3 lsb">\u2212<span class="fff2 ls1aa ws14b">207<span class="_30 blank"></span>1<span class="_8 blank"></span>2</span></span></div><div class="t m0 x35 ha y37b fff2 fs5 fc3 sc0 ls1bb ws15d">24<span class="_15 blank"></span><span class="fff3 lsb">\u2212<span class="fff2 ws15a">10<span class="_31 blank"> </span>6 12 28</span></span></div><div class="t m0 x35 ha y37c fff2 fs5 fc3 sc0 ls1bb ws15d">24<span class="_d blank"></span><span class="fff3 lsb">\u2212<span class="fff2 lsc2 wsd4">56<span class="_15 blank"></span><span class="fff3 lsb">\u2212<span class="fff2 lsdb">5</span>\u2212<span class="fff2">1</span></span></span></span></div><div class="t m0 xc3 h9 y377 fff1 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc3 h9 y378 fff1 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc3 h9 y379 fff1 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xa7 ha y37d fff4 fs5 fc3 sc0 ls1b4">0<span class="fff5 lsb ws24">x</span><span class="fsc ls4d v9">1</span><span class="fff6 ls4">+</span>0<span class="fff5 ls1b5">x</span><span class="fsc ls175 v9">2</span><span class="fff6 ls4">+</span>0<span class="fff5 ls1b5">x</span><span class="fsc ls53 v9">3</span><span class="fff6 ls6">=</span><span class="lsb">1</span></div><div class="t m0 x13 ha y37e fff7 fs5 fc3 sc0 lsb ws24">x<span class="fff8 fsc ls51 v9">1</span><span class="fff9 ls75 ws25">=\u2212<span class="_d blank"></span><span class="fff8 ls17c">2<span class="fff9 ls4">\u2212</span><span class="lsb">6<span class="fff7 ls3">s<span class="fff9 ls4">\u2212</span></span><span class="ls2">4<span class="fff7 ls54 ws15e">t, x</span></span></span></span></span></div><div class="t m0 xa7 h86 y37f fff8 fsc fc3 sc0 ls55">2<span class="fff9 fs5 ls6 va">=<span class="fff7 ls56 ws15f">s, x</span></span></div><div class="t m0 xc4 h87 y37f fff8 fsc fc3 sc0 ls51">3<span class="fff9 fs5 ls6 va">=<span class="fff8 ls4">1<span class="fff9">\u2212</span><span class="lsb">3<span class="fff7 ls54 ws15e">t, x</span></span></span></span></div><div class="t m0 x4 h87 y37f fff8 fsc fc3 sc0 ls174">4<span class="fff9 fs5 ls6 va">=<span class="fff8 ls17c">2<span class="fff9 ls4">\u2212</span><span class="lsb">5<span class="fff7 ls54 ws15e">t, x</span></span></span></span></div><div class="t m0 xc5 h86 y37f fff8 fsc fc3 sc0 ls51">5<span class="fff9 fs5 ls6 va">=<span class="fff7 lsb">t</span></span></div><div class="t m0 x3f h38 ya3 ff4 fs10 fc2 sc0 ls7d ws9a">Capítulo 1 - Sistemas de Equações Linear<span class="_0 blank"></span>es e Matrizes<span class="_12 blank"> </span><span class="ff1 fsf fc1 ls7c ws7f v14">\u2022 \u2022 \u2022<span class="_12 blank"> </span><span class="fs7 lsd v5">33</span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y380 we h88" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg8.png"><div class="t m0 xb h4 y381 ff3 fs2 fc2 sc0 lsa ws191">Simple Proof,\u201d de Thomas <span class="_0 blank"></span>Y<span class="_5 blank"></span>uster<span class="_0 blank"></span>, <span class="ff4">Mathematics Magazine</span><span class="lsb ws192">, V<span class="_32 blank"></span>ol.</span></div><div class="t m0 xb h4 y382 ff3 fs2 fc2 sc0 lsb ws193">57, No. 2, 1984, páginas 93-94.) Por outro lado, <span class="ff4 lsa">uma forma</span></div><div class="t m0 xb h4 y383 ff4 fs2 fc2 sc0 lsb ws194">escalonada de uma dada matriz não é única<span class="ff3 lsa ws195">: diferentes seqüên-</span></div><div class="t m0 xb h4 y384 ff3 fs2 fc2 sc0 lsa ws196">cias de operações sobre linhas podem produzir formas escalo-</div><div class="t m0 xb h4 y385 ff3 fs2 fc2 sc0 lsa ws6">nadas diferentes.</div><div class="t m0 xb h4 y386 ff3 fs2 fc2 sc0 lsa ws6">Resolva por eliminação de Gauss-Jordan.</div><div class="t m0 xb h35 y387 ff4 fs2 fc1 sc0 lsb">Solução.</div><div class="t m0 xb h4 y388 ff3 fs2 fc2 sc0 lsb ws3c">A<span class="_25 blank"> </span>matriz aumentada do sistema é</div><div class="t m0 x14 h4 y389 ff3 fs2 fc2 sc0 lsb ws197">Somando \u20132 vezes a primeira linha à segunda e quarta linhas dá</div><div class="t m0 x14 h4 y38a ff3 fs2 fc2 sc0 lsb ws198">Multiplicando a segunda linha por \u20131 e depois somando \u20135</div><div class="t m0 x14 h4 y38b ff3 fs2 fc2 sc0 lsb ws40">vezes a nova segunda linha à terceira linha e \u20134 vezes a nova</div><div class="t m0 x14 h4 y38c ff3 fs2 fc2 sc0 lsb ws3c">segunda linha à quarta linha dá</div><div class="t m0 xc6 h9 y38d fffa fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xc6 h9 y38e fffa fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y38f fffa fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y390 fffa fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y391 fffa fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xbd ha y392 fffb fs5 fc3 sc0 lsbe ws86">13<span class="_15 blank"></span><span class="fffc lsb">\u2212<span class="fffb ls1a0">20200</span></span></div><div class="t m0 xbd ha y393 fffb fs5 fc3 sc0 lsbc ws18b">0012031</div><div class="t m0 xbd ha y394 fffb fs5 fc3 sc0 ls1bc ws18c">0000000</div><div class="t m0 xbd ha y395 fffb fs5 fc3 sc0 ls1bd ws18d">0000062</div><div class="t m0 xc7 h9 y38d fffa fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc7 h9 y38e fffa fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y38f fffa fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y390 fffa fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y391 fffa fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xc6 h9 y396 fffd fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xc6 h9 y397 fffd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y398 fffd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y399 fffd fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xc6 h9 y39a fffd fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xbd ha y39b fffe fs5 fc3 sc0 lsbe ws86">13<span class="_15 blank"></span><span class="ffff lsb">\u2212<span class="fffe ls1c7">20200</span></span></div><div class="t m0 xbd ha y39c fffe fs5 fc3 sc0 lsbe ws86">00<span class="_15 blank"></span><span class="ffff lsb">\u2212<span class="fffe lsdb">1</span>\u2212<span class="fffe ls186 ws115">20<span class="_15 blank"></span><span class="ffff lsb">\u2212<span class="fffe lsdb">3</span>\u2212<span class="fffe">1</span></span></span></span></div><div class="t m0 xbd ha y39d fffe fs5 fc3 sc0 lsb ws18e">0 0 5<span class="_33 blank"> </span>10 0<span class="_33 blank"> </span>15 5</div><div class="t m0 xbd ha y39e fffe fs5 fc3 sc0 lsbe ws86">00480<span class="_30 blank"></span>1<span class="_8 blank"></span>86</div><div class="t m0 xc7 h9 y396 fffd fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc7 h9 y397 fffd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y398 fffd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y399 fffd fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc7 h9 y39a fffd fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x17 h9 y39f ff100 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x17 h9 y3a0 ff100 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y3a1 ff100 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y3a2 ff100 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y3a3 ff100 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x56 ha y3a4 ff101 fs5 fc3 sc0 lsbe ws86">13<span class="_15 blank"></span><span class="ff102 lsb">\u2212<span class="ff101 ls1c7">20200</span></span></div><div class="t m0 x56 ha y3a5 ff101 fs5 fc3 sc0 ls1b8 ws156">26<span class="_15 blank"></span><span class="ff102 lsb">\u2212<span class="ff101 lsdb">5</span>\u2212<span class="ff101 ls1b9 ws159">24<span class="_15 blank"></span><span class="ff102 lsb">\u2212<span class="ff101 lsdb">3</span>\u2212<span class="ff101">1</span></span></span></span></div><div class="t m0 x56 ha y3a6 ff101 fs5 fc3 sc0 lsb ws18e">0 0 5<span class="_33 blank"> </span>10 0<span class="_33 blank"> </span>15 5</div><div class="t m0 x56 ha y3a7 ff101 fs5 fc3 sc0 ls187 ws18f">26084<span class="_30 blank"></span>1<span class="_8 blank"></span>86</div><div class="t m0 xc8 h9 y39f ff100 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xc8 h9 y3a0 ff100 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc8 h9 y3a1 ff100 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc8 h9 y3a2 ff100 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xc8 h9 y3a3 ff100 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x71 ha y3a8 ff103 fs5 fc3 sc0 lsb ws24">x<span class="ff104 fsc ls51 v9">1</span><span class="ff105 ls6">+</span><span class="ff104">3</span>x<span class="ff104 fsc ls8f v9">2</span><span class="ff105 ls6">\u2212<span class="ff104 ls1">2</span></span>x<span class="ff104 fsc ls1be v9">3</span><span class="ff105 ls6">+<span class="ff104 ls1">2</span></span>x<span class="ff104 fsc ls1bf v9">5</span><span class="ff105 ls1c0">=</span><span class="ff104">0</span></div><div class="t m0 x51 h1a y3a9 ff104 fs5 fc3 sc0 ls1">2<span class="ff103 lsb ws24">x</span><span class="fsc ls51 v9">1</span><span class="ff105 ls6 v0">+<span class="ff104 lsb">6<span class="ff103 ws24">x</span><span class="fsc ls8f v9">2</span></span><span class="ls8e">\u2212<span class="ff104 lsb">5<span class="ff103 ws24">x</span><span class="fsc ls51 v9">3</span></span><span class="ls1c1">\u2212</span></span></span><span class="v0">2<span class="ff103 lsb ws24">x</span><span class="fsc ls6a v9">4</span><span class="ff105 ls1c2">+</span><span class="ls2">4<span class="ff103 lsb ws24">x</span><span class="fsc ls51 v9">5</span><span class="ff105 lsd5">\u2212</span><span class="lsb">3<span class="ff103 ws24">x</span><span class="fsc ls53 v9">6</span><span class="ff105 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff104 lsb">1</span></span></span></span></span></div><div class="t m0 xc9 h1a y3aa ff104 fs5 fc3 sc0 lsb">5<span class="ff103 ws24">x</span><span class="fsc ls51 v9">3</span><span class="ff105 ls6 v0">+</span><span class="ws24 v0">10<span class="ff103">x</span><span class="fsc ls1c3 v9">4</span><span class="ff105 ls6">+</span>15<span class="ff103">x</span><span class="fsc ls53 v9">6</span><span class="ff105 ls178">=</span>5</span></div><div class="t m0 x51 h1a y3ab ff104 fs5 fc3 sc0 ls1">2<span class="ff103 lsb ws24">x</span><span class="fsc ls51 v9">1</span><span class="ff105 ls6 v0">+<span class="ff104 lsb">6<span class="ff103 ws24">x</span><span class="fsc ls1c4 v9">2</span></span><span class="ls1c5">+<span class="ff104 lsb">8<span class="ff103 ws24">x</span><span class="fsc ls6a v9">4</span></span><span class="ls1c2">+<span class="ff104 ls2">4<span class="ff103 lsb ws24">x</span><span class="fsc ls51 v9">5</span></span></span></span>+<span class="ff104 lsb">18<span class="ff103 ws24">x</span><span class="fsc ls53 v9">6</span></span><span class="ls178">=<span class="ff104 lsb">6</span></span></span></div><div class="t m0 x2 he y3ac ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 4<span class="_34 blank"> </span><span class="fs1 fc2 ls7a ws7e">Eliminação de Gauss-Jordan</span></div><div class="t m0 xb h2a ya3 ff1 fs7 fc1 sc0 lsd ws2e">34 <span class="fsf ls7c ws7f vf">\u2022 \u2022 \u2022<span class="_12 blank"> </span></span><span class="ff3 fs10 fc2 ls7d ws80">Álgebra Linear com <span class="_0 blank"></span>Aplicações</span></div><div class="t m0 xb h89 y3ad ff2 fs2 fc2 sc0 lsa ws6">Karl Friedrich Gauss</div><div class="t m0 xb h89 y3ae ff2 fs2 fc2 sc0 lsa ws6">Wilhelm Jordan</div><div class="t m0 xca h2c y3af ff2 fs10 fc2 sc0 ls7d ws199">Karl Friedrich Gauss <span class="ff3">(1777\u20131855) foi um</span></div><div class="t m0 xca h2c y3b0 ff3 fs10 fc2 sc0 lsb ws19a">cientista e matemático alemão. Às vezes chama-</div><div class="t m0 xca h2c y3b1 ff3 fs10 fc2 sc0 lsb ws35">do \u201cpríncipe dos matemáticos,\u201d Gauss é coloca-</div><div class="t m0 xca h2c y3b2 ff3 fs10 fc2 sc0 ls7d ws19b">do junto a Newton e <span class="_0 blank"></span>Arquimedes como um dos</div><div class="t m0 xca h2c y3b3 ff3 fs10 fc2 sc0 ls7d ws19c">três maiores matemáticos de todos os tempos.</div><div class="t m0 xca h2c y3b4 ff3 fs10 fc2 sc0 lsb ws19d">Em toda a história da Matemática, talvez nunca</div><div class="t m0 xca h2c y3b5 ff3 fs10 fc2 sc0 lsb ws19e">tenha havido uma criança tão precoce como</div><div class="t m0 xca h2c y3b6 ff3 fs10 fc2 sc0 ls7d ws19f">Gauss\u2014por sua própria iniciativa desenvolveu</div><div class="t m0 xca h2c y3b7 ff3 fs10 fc2 sc0 ls7d ws1a0">os rudimentos da aritmética antes mesmo de</div><div class="t m0 xca h2c y3b8 ff3 fs10 fc2 sc0 lsb ws1a1">começar a falar<span class="_0 blank"></span>. Certo dia, antes de completar</div><div class="t m0 xca h2c y3b9 ff3 fs10 fc2 sc0 ls7d ws1a2">três anos, seu gênio se mostrou a seus pais de</div><div class="t m0 xca h2c y3ba ff3 fs10 fc2 sc0 lsb ws1a3">uma maneira muito dramática. Seu pai estava</div><div class="t m0 xca h2c y3bb ff3 fs10 fc2 sc0 lsb ws1a4">preparando a folha de pagamento semanal dos</div><div class="t m0 xca h2c y3bc ff3 fs10 fc2 sc0 ls7d ws1a5">trabalhadores sob seu encargo enquanto o garo-</div><div class="t m0 xca h2c y3bd ff3 fs10 fc2 sc0 lsb ws1a6">to observava quieto desde um canto. Quando seu</div><div class="t m0 xca h2c y3be ff3 fs10 fc2 sc0 lsb ws1a7">pai terminou a longa e tediosa conta, Gauss</div><div class="t m0 xca h2c y3bf ff3 fs10 fc2 sc0 ls7d ws1a8">informou-lhe que havia um erro no resultado e</div><div class="t m0 xca h2c y3c0 ff3 fs10 fc2 sc0 lsb ws1a9">forneceu a resposta que ele havia calculado de</div><div class="t m0 xca h2c y3c1 ff3 fs10 fc2 sc0 ls7d ws1aa">cabeça. Para grande surpresa dos pais, uma ver-</div><div class="t m0 xca h2c y3c2 ff3 fs10 fc2 sc0 ls7d ws1ab">ificação da conta mostrou que Gauss estava</div><div class="t m0 xca h2c y3c3 ff3 fs10 fc2 sc0 ls7d">certo!</div><div class="t m0 x87 h2c y3c4 ff3 fs10 fc2 sc0 ls7d ws1ac">Em sua tese de Doutorado, Gauss deu a</div><div class="t m0 xca h2c y3c5 ff3 fs10 fc2 sc0 ls1c6 ws1ad">primeira prova completa do T<span class="_5 blank"></span>eorema</div><div class="t m0 xca h2c y3c6 ff3 fs10 fc2 sc0 lsb ws1ae">Fundamental da Álgebra, que afirma que o</div><div class="t m0 xca h2c y3c7 ff3 fs10 fc2 sc0 lsb ws1af">número de soluções de cada equação polinomial</div><div class="t m0 xca h2c y3c8 ff3 fs10 fc2 sc0 lsb ws1b0">coincide com o seu grau. <span class="_0 blank"></span>Aos 19 anos de idade,</div><div class="t m0 xca h2c y3c9 ff3 fs10 fc2 sc0 ls7d ws1b1">resolveu um problema que frustrou Euclides,</div><div class="t m0 xca h2c y3ca ff3 fs10 fc2 sc0 lsb ws1b2">inscrevendo um polígono regular de dezessete</div><div class="t m0 xca h2c y3cb ff3 fs10 fc2 sc0 lsb ws1b3">lados em um círculo usando somente régua e</div><div class="t m0 xca h2c y3cc ff3 fs10 fc2 sc0 lsb ws1b4">compasso; e em 1801, aos 24 anos, ele publicou</div><div class="t m0 xca h2c y3cd ff3 fs10 fc2 sc0 ls1c8 ws1b5">sua primeira obra-prima, <span class="ff4">Disquisi-tiones</span></div><div class="t m0 xca h2c y3ce ff4 fs10 fc2 sc0 ls7d">Arithmeticae<span class="ff3 lsb ws1b6">, considerada por muitos como uma</span></div><div class="t m0 xca h2c y3cf ff3 fs10 fc2 sc0 ls7d ws1b7">das mais brilhantes realizações matemáticas.</div><div class="t m0 xca h2c y3d0 ff3 fs10 fc2 sc0 ls7d ws1b8">Neste trabalho, Gauss sistematizou o estudo da</div><div class="t m0 xca h2c y3d1 ff3 fs10 fc2 sc0 ls7d ws1b9">T<span class="_0 blank"></span>eoria de Números (propriedades dos inteiros) e</div><div class="t m0 xca h2c y3d2 ff3 fs10 fc2 sc0 ls7d ws1ba">formulou os conceitos básicos que constituem o</div><div class="t m0 xca h2c y3d3 ff3 fs10 fc2 sc0 ls7d ws9a">fundamento deste assunto.</div><div class="t m0 xa2 h2c y3d4 ff3 fs10 fc2 sc0 ls1c9 ws1bb">Entre suas inúmeras realizações, Gauss</div><div class="t m0 x56 h2c y3d5 ff3 fs10 fc2 sc0 ls7d ws1bc">descobriu a assim chamada curva de Gauss, ou</div><div class="t m0 x56 h2c y3d6 ff3 fs10 fc2 sc0 ls1ca ws1bd">em forma de sino, que é fundamental em</div><div class="t m0 x56 h2c y3d7 ff3 fs10 fc2 sc0 ls1cb ws1be">Probabilidade, deu a primeira interpretação</div><div class="t m0 x56 h2c y3d8 ff3 fs10 fc2 sc0 lsb ws1bf">geométrica dos números complexos e estabele-</div><div class="t m0 x56 h2c y3d9 ff3 fs10 fc2 sc0 ls1cc ws1c0">ceu seu papel fundamental na Matemática,</div><div class="t m0 x56 h2c y3da ff3 fs10 fc2 sc0 ls7d ws1c1">desenvolveu métodos para caracterizar superfí-</div><div class="t m0 x56 h2c y3db ff3 fs10 fc2 sc0 ls7d ws1c2">cies de maneira intrínseca por meio de curvas</div><div class="t m0 x56 h2c y3dc ff3 fs10 fc2 sc0 ls7d ws1c3">nestas superfícies, desenvolveu a teoria das apli-</div><div class="t m0 x56 h2c y3dd ff3 fs10 fc2 sc0 ls7d ws1c4">cações conformes (as que preservam ângulo) e</div><div class="t m0 x56 h2c y3de ff3 fs10 fc2 sc0 lsb ws1c5">descobriu a geometria não-euclidiana 30 anos</div><div class="t m0 x56 h2c y3df ff3 fs10 fc2 sc0 ls7d ws1c6">antes destas idéias serem publicadas por outros.</div><div class="t m0 x56 h2c y3e0 ff3 fs10 fc2 sc0 ls7d ws1c7">Na Física, ele deu contribuições relevantes à</div><div class="t m0 x56 h2c y3e1 ff3 fs10 fc2 sc0 lsb ws1c8">teoria de lentes e à ação capilar e, com W<span class="_0 blank"></span>ilhelm</div><div class="t m0 x56 h2c y3e2 ff3 fs10 fc2 sc0 lsb ws1c9">W<span class="_5 blank"></span>eber, realizou trabalho fundamental em eletro-</div><div class="t m0 x56 h2c y3e3 ff3 fs10 fc2 sc0 lsb ws1ca">magnetismo. Gauss inventou o heliotrópio, o</div><div class="t m0 x56 h2c y3e4 ff3 fs10 fc2 sc0 lsb ws9a">magnetômetro bifilar e um eletrotelégrafo.</div><div class="t m0 xa2 h2c y3e5 ff3 fs10 fc2 sc0 ls7d ws1cb">Gauss era profundamente religioso e aris-</div><div class="t m0 x56 h2c y3e6 ff3 fs10 fc2 sc0 lsb ws1cc">tocrático em sua conduta. Ele dominava línguas</div><div class="t m0 x56 h2c y3e7 ff3 fs10 fc2 sc0 lsb ws1cd">estrangeiras com facilidade, lia extensivamente</div><div class="t m0 x56 h2c y3e8 ff3 fs10 fc2 sc0 lsb ws1ce">e apreciava mineralogia e botânica como <span class="ff4">hobby</span>.</div><div class="t m0 x56 h2c y3e9 ff3 fs10 fc2 sc0 lsb ws1cf">Ele não gostava de lecionar e era em geral frio e</div><div class="t m0 x56 h2c y3ea ff3 fs10 fc2 sc0 ls7d ws1d0">desencorajador com outros matemáticos, pos-</div><div class="t m0 x56 h2c y3eb ff3 fs10 fc2 sc0 lsb ws1d1">sivelmente por que ele já havia antecipado seus</div><div class="t m0 x56 h2c y3ec ff3 fs10 fc2 sc0 ls7d ws1d2">trabalhos. Diz-se que se Gauss tivesse publicado</div><div class="t m0 x56 h2c y3ed ff3 fs10 fc2 sc0 ls1c8 ws1d3">todos suas descobertas, o estado atual da</div><div class="t m0 x56 h2c y3ee ff3 fs10 fc2 sc0 ls7d ws99">Matemática estaria 50 anos à frente. Ele foi, sem</div><div class="t m0 x56 h2c y3ef ff3 fs10 fc2 sc0 lsb ws9a">duvida, o maior matemático da era moderna.</div><div class="t m0 x56 h2c y3f0 ff2 fs10 fc2 sc0 ls7d ws1d4">Wilhelm Jordan <span class="ff3">(1842\u20131899) foi um enge-</span></div><div class="t m0 x56 h2c y3f1 ff3 fs10 fc2 sc0 lsb ws1c4">nheiro alemão especializado em Geodesia. Sua</div><div class="t m0 x56 h2c y3f2 ff3 fs10 fc2 sc0 ls7d ws1d5">contribuição à resolução de sistemas lineares</div><div class="t m0 x56 h2c y3f3 ff3 fs10 fc2 sc0 lsb ws1d6">apareceu em seu livro popular<span class="_0 blank"></span>, <span class="ff4 ls7d ws1d7">Handbuch der</span></div><div class="t m0 x56 h2c y3f4 ff4 fs10 fc2 sc0 ls7d ws190">V<span class="_5 blank"></span>ermessungskunde <span class="ff3 ws1d8">(Manual da Geodesia) em</span></div><div class="t m0 x56 h2c y3f5 ff3 fs10 fc2 sc0 lsb">1888.</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y25 w9 h59" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bg9.png"><div class="t m0 xb h4 ya5 ff3 fs2 fc2 sc0 lsb ws1df">Permutando as terceira e quarta linhas e então multiplicando a</div><div class="t m0 xb h4 y2c8 ff3 fs2 fc2 sc0 lsa ws6">terceira linha da matriz resultante por <span class="_20 blank"> </span>dá a forma escalonada</div><div class="t m0 xb h4 y3f6 ff3 fs2 fc2 sc0 lsb ws1e0">Somando \u20133 vezes a terceira linha à segunda linha e depois</div><div class="t m0 xb h4 y3f7 ff3 fs2 fc2 sc0 lsb ws1e1">somando 2 vezes a segunda linha da matriz resultante à primeira</div><div class="t m0 xb h4 y3f8 ff3 fs2 fc2 sc0 lsa ws6">linha fornece a forma escalonada reduzida por linhas</div><div class="t m0 xb h4 y3f9 ff3 fs2 fc2 sc0 lsa ws6">O sistema de equações correspondente é</div><div class="t m0 xb h8a y3fa ff3 fs2 fc2 sc0 lsa ws1e2">(Nós descartamos a última equação, 0<span class="ff4 lsb ws17">x</span><span class="fs3 ls1cd v6">1</span><span class="lsb ws1e3 v0">+ 0<span class="ff4 ws17">x</span><span class="fs3 ls1cd v6">2</span>+ 0<span class="ff4 ws17">x</span><span class="fs3 ls1ce v6">3</span>+ 0<span class="ff4 ws17">x</span><span class="fs3 ws7 v6">4 </span>+ 0<span class="ff4 ws17">x</span><span class="fs3 v6">5</span></span></div><div class="t m0 xb h4 y3fb ff3 fs2 fc2 sc0 lsb ws1e4">+ 0<span class="ff4 ws17">x</span><span class="fs3 ls1cf v6">6</span>= 0, pois ela é automaticamente satisfeita pelas soluções</div><div class="t m0 xb h4 y3fc ff3 fs2 fc2 sc0 lsa ws1e5">das demais equações.) Resolvendo as variáveis líderes, obtemos</div><div class="t m0 xb h4 y3fd ff3 fs2 fc2 sc0 lsa ws1e6">Se dermos os valores arbitrários <span class="ff4 lsb">r</span><span class="ls1d0">,<span class="ff4 ls1d1">s<span class="ff3">e</span>t</span></span>às variáveis livres <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws16 v6">2</span><span class="ff3 ws7 v0">, </span><span class="v0">x<span class="ff3 fs3 v6">4</span></span></span></div><div class="t m0 xb h4 y3fe ff3 fs2 fc2 sc0 ls30">e<span class="ff4 lsb ws17">x</span><span class="fs3 lsb ws16 v6">5</span><span class="lsa ws1e7">, respectivamente, então a solução geral é dada pelas fórmu-</span></div><div class="t m0 xb h4 y3ff ff3 fs2 fc2 sc0 lsc">las</div><div class="t m0 x95 h14 y400 ffe fs2 fc1 sc0 lsb">®</div><div class="t m0 xb he y401 ff1 fs7 fc1 sc0 ls71 ws7">Retro-substituição <span class="ff3 fs2 fc2 lsa ws1e8">Às vezes é preferível resolver um sis-</span></div><div class="t m0 xb h4 y402 ff3 fs2 fc2 sc0 lsb ws1e9">tema de equações lineares por eliminação gaussiana para levar</div><div class="t m0 xb h4 y403 ff3 fs2 fc2 sc0 lsb ws1ea">a matriz aumentada à forma escalonada sem continuar até</div><div class="t m0 xb h4 y404 ff3 fs2 fc2 sc0 lsa ws1eb">chegar à forma escalonada reduzida por linhas. Quando isto é</div><div class="t m0 xb h4 y405 ff3 fs2 fc2 sc0 lsa ws1ec">feito, o correspondente sistema de equações pode ser resolvido</div><div class="t m0 xb h4 y406 ff3 fs2 fc2 sc0 lsc ws1ed">por uma técnica chamada retro-substituição. O próximo exemp-</div><div class="t m0 xb h4 y407 ff3 fs2 fc2 sc0 lsb ws3c">lo ilustra esta idéia.</div><div class="t m0 xb h4 y408 ff3 fs2 fc2 sc0 lsb ws1ee">Das contas do Exemplo 4, uma forma escalonada da matriz</div><div class="t m0 xb h4 y409 ff3 fs2 fc2 sc0 lsb ws6">aumentada é</div><div class="t m0 xb h4 y40a ff3 fs2 fc2 sc0 lsa ws6">Para resolver o sistema de equações correspondente</div><div class="t m0 xb h4 y40b ff3 fs2 fc2 sc0 lsa ws6">nós procedemos da seguinte maneira:</div><div class="t m0 x14 h4 y40c ff2 fs2 fc2 sc0 lsa ws6">Passo 1. <span class="ff3">Resolva as equações para as variáveis líderes.</span></div><div class="t m0 x14 h4 y40d ff2 fs2 fc2 sc0 lsa ws1ef">Passo 2. <span class="ff3 lsb ws1f0">Começando com a equação de baixo e trabalhando</span></div><div class="t m0 x85 h4 y40e ff3 fs2 fc2 sc0 lsb ws1f1">para cima, substitua sucessivamente cada equação em todas</div><div class="t m0 x85 h4 y40f ff3 fs2 fc2 sc0 lsb ws3c">as equações acima dela.</div><div class="t m0 x85 h4 y410 ff3 fs2 fc2 sc0 lsb ws3c">Substituindo <span class="_35 blank"> </span>na segunda equação dá</div><div class="t m0 x14 h4 y411 ff3 fs2 fc2 sc0 lsa ws7">Substituindo <span class="_4 blank"> </span><span class="ff4 ls167">x</span><span class="fs3 ls2c v6">3</span><span class="lsb ws3c v0">= \u20132<span class="ff4 ws17">x</span><span class="fs3 ls2c v6">4</span>na primeira equação, dá</span></div><div class="t m0 x14 h4 y412 ff2 fs2 fc2 sc0 lsa ws1f2">Passo 3.<span class="_14 blank"> </span><span class="ff3">Atribua valores arbitrários às variáveis livres, se hou-</span></div><div class="t m0 x85 h4 y413 ff3 fs2 fc2 sc0 lsc ws1d9">ver<span class="_0 blank"></span>.</div><div class="t m0 x15 h4 y414 ff3 fs2 fc2 sc0 lsa ws1f3">Atribuindo os valores arbitrários <span class="ff4 lsb">r</span><span class="ls1d2">,<span class="ff4 ls15">s<span class="ff3">e</span>t</span><span class="lsb ws7">a <span class="ff4 ws17">x</span><span class="fs3 ws16 v6">2</span><span class="v0">, <span class="ff4 ws17">x</span><span class="fs3 ls1d3 v6">4</span><span class="ls15">e</span><span class="ff4 ws17">x</span><span class="fs3 ws16 v6">5</span></span></span></span><span class="v0">, respec-</span></div><div class="t m0 x14 h4 y415 ff3 fs2 fc2 sc0 lsb ws3c">tivamente, a solução geral é dada pelas fórmulas</div><div class="t m0 x14 h14 y416 ff3 fs2 fc2 sc0 lsb ws3c">Isto confere com a solução obtida no Exemplo 4.<span class="_36 blank"> </span><span class="ffe fc1">®</span></div><div class="t m0 x14 h4 y417 ff3 fs11 fc1 sc0 ls1b6 ws14a">OBSER<span class="_0 blank"></span>V<span class="_5 blank"></span>AÇÃO<span class="fs2 ls1d4">.<span class="fc2 lsa ws1f4">Os valores arbitrários que atribuímos às variáveis</span></span></div><div class="t m0 x14 h4 y418 ff3 fs2 fc2 sc0 lsa ws1f5">livres são, muitas vezes, chamados <span class="ffd">parâmetros</span>. Nós geralmente</div><div class="t m0 x14 h4 y419 ff3 fs2 fc2 sc0 lsa ws1f6">usamos as letras <span class="ff4 lsb">r</span><span class="ls1d5">,<span class="ff4 lsb">s</span>,<span class="ff4 lsb">t</span></span>,... para os parâmetros, mas também</div><div class="t m0 x14 h4 y41a ff3 fs2 fc2 sc0 lsa ws1df">podem ser usadas quaisquer letras que não entrem em conflito</div><div class="t m0 x14 h4 y41b ff3 fs2 fc2 sc0 lsa ws6">com as variáveis.</div><div class="t m0 x14 h4 y41c ff3 fs2 fc2 sc0 lsa">Resolva</div><div class="t m0 x14 h4 y41d ff3 fs2 fc2 sc0 lsa ws6">por eliminação gaussiana e retro-substituição.</div><div class="t m0 x14 h35 y41e ff4 fs2 fc1 sc0 lsb">Solução.</div><div class="t m0 x14 h4 y41f ff3 fs2 fc2 sc0 lsb ws45">Este é o sistema do Exemplo 3 da Seção 1.1. Naquele exemplo,</div><div class="t m0 x14 h4 y420 ff3 fs2 fc2 sc0 lsb ws3c">nós convertemos a matriz aumentada</div><div class="t m0 x14 h4 y421 ff3 fs2 fc2 sc0 lsa ws3c">à forma escalonada</div><div class="t m0 x14 h4 y422 ff3 fs2 fc2 sc0 lsa ws3c">O sistema correspondente a esta matriz é</div><div class="t m0 xcb h9 y423 ff106 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xcb h9 y424 ff106 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xcb h9 y425 ff106 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x42 ha y426 ff107 fs5 fc3 sc0 lsbe ws1da">11 2<span class="_27 blank"> </span>9</div><div class="t m0 x42 h8b y427 ff107 fs5 fc3 sc0 lsc2 wsd4">01<span class="_15 blank"></span><span class="ff108 lsd1">\u2212<span class="ff107 fsc lsb v5">7</span></span></div><div class="t m0 xb6 h3c y428 ff107 fsc fc3 sc0 lsca">2<span class="ff108 fs5 lscb v5">\u2212</span><span class="lsb vb">17</span></div><div class="t m0 xb8 h27 y428 ff107 fsc fc3 sc0 lsb">2</div><div class="t m0 x42 ha y429 ff107 fs5 fc3 sc0 lsbe ws1db">00 1<span class="_2 blank"> </span>3</div><div class="t m0 x5e h9 y42a ff106 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x5e h9 y42b ff106 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x5e h9 y42c ff106 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xcb h9 y42d ff109 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xcb h9 y42e ff109 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xcb h9 y42f ff109 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x9b ha y430 ff10a fs5 fc3 sc0 lsbc">1129</div><div class="t m0 x9b ha y431 ff10a fs5 fc3 sc0 lsbd ws85">24<span class="_15 blank"></span><span class="ff10b lsb">\u2212<span class="ff10a lsbe">31</span></span></div><div class="t m0 x9b ha y432 ff10a fs5 fc3 sc0 lsbe ws86">36<span class="_15 blank"></span><span class="ff10b lsb">\u2212<span class="ff10a lsbf">50</span></span></div><div class="t m0 x24 h9 y42d ff109 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x24 h9 y42e ff109 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x24 h9 y42f ff109 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x29 ha y433 ff10c fs5 fc3 sc0 ls5">x<span class="ff10d ls8b">+</span>y<span class="ff10d ls6">+<span class="ff10e ls1">2</span></span><span class="ls8c">z<span class="ff10d ls6">=<span class="ff10e lsb">9</span></span></span></div><div class="t m0 x42 ha y434 ff10e fs5 fc3 sc0 ls1">2<span class="ff10c ls5">x<span class="ff10d ls6">+</span></span><span class="ls2">4<span class="ff10c ls5">y<span class="ff10d ls6">\u2212</span></span><span class="lsb">3<span class="ff10c ls8d">z<span class="ff10d ls6">=</span></span>1</span></span></div><div class="t m0 x42 ha y435 ff10e fs5 fc3 sc0 lsb">3<span class="ff10c ls5">x<span class="ff10d ls4b">+</span></span>6<span class="ff10c ls5">y<span class="ff10d ls6">\u2212</span></span>5<span class="ff10c ls8d">z<span class="ff10d ls6">=</span></span>0</div><div class="t m0 x3f he y436 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 6<span class="_34 blank"> </span><span class="fs1 fc2 ls7a ws7e">Eliminação Gaussiana</span></div><div class="t m0 xa1 ha y437 ff10f fs5 fc3 sc0 lsb ws24">x<span class="ff110 fsc ls53 v9">1</span><span class="ff111 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff110 lsb">3<span class="ff10f ls1d6">r<span class="ff111 ls4">\u2212</span></span><span class="ls2">4<span class="ff10f ls3">s<span class="ff111 ls4">\u2212</span></span><span class="ls1">2<span class="ff10f ls54 ws15e">t, x</span></span></span></span></span></div><div class="t m0 xcb h69 y438 ff110 fsc fc3 sc0 ls55">2<span class="ff111 fs5 ls6 va">=<span class="ff10f ls1e3 ws1dc">r,<span class="_37 blank"> </span>x</span></span><span class="ls53">3<span class="ff111 fs5 ls75 ws25 va">=\u2212<span class="_d blank"></span><span class="ff110 ls1">2<span class="ff10f ls56 ws15f">s, x</span></span></span></span></div><div class="t m0 x3d h8c y438 ff110 fsc fc3 sc0 ls6a">4<span class="ff111 fs5 ls6 va">=<span class="ff10f ls56 ws15f">s, x</span></span></div><div class="t m0 x2d h8c y438 ff110 fsc fc3 sc0 ls53">5<span class="ff111 fs5 ls6 va">=<span class="ff10f ls54 ws15e">t, x</span></span></div><div class="t m0 xcc h8d y438 ff110 fsc fc3 sc0 ls53">6<span class="ff111 fs5 ls1d7 va">=</span><span class="lsb v23">1</span></div><div class="t m0 xa3 h27 y439 ff110 fsc fc3 sc0 lsb">3</div><div class="t m0 xa2 ha y43a ff112 fs5 fc3 sc0 ls36">x<span class="ff113 fsc ls53 v9">1</span><span class="ff114 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff113 lsb">3<span class="ff112 ws24">x</span><span class="fsc ls175 v9">2</span><span class="ff114 ls4">\u2212</span><span class="ls2">4<span class="ff112 ls176">x</span><span class="fsc ls1d8 v9">4</span><span class="ff114 ls4">\u2212</span><span class="ls1">2<span class="ff112 ls69">x</span></span></span><span class="fsc v9">5</span></span></span></div><div class="t m0 xa2 ha y43b ff112 fs5 fc3 sc0 ls36">x<span class="ff113 fsc ls53 v9">3</span><span class="ff114 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff113 ls1">2<span class="ff112 lsb ws24">x<span class="ff113 fsc v9">4</span></span></span></span></div><div class="t m0 xa2 h8e y43c ff112 fs5 fc3 sc0 ls36">x<span class="ff113 fsc ls53 v9">6</span><span class="ff114 ls1d7">=<span class="ff113 fsc lsb v5">1</span></span></div><div class="t m0 x28 h27 y43d ff113 fsc fc3 sc0 lsb">3</div><div class="t m0 xcb ha y43e ff115 fs5 fc3 sc0 ls36">x<span class="ff116 fsc ls53 v9">1</span><span class="ff117 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff116 lsb">3<span class="ff115 ws24">x</span><span class="fsc ls175 v9">2</span><span class="ff117 ls4">+</span><span class="ls1">2<span class="ff115 ls69">x</span><span class="fsc ls4d v9">3</span><span class="ff117 ls4">\u2212</span>2<span class="ff115 ls69">x</span></span><span class="fsc v9">5</span></span></span></div><div class="t m0 xcb ha y43f ff115 fs5 fc3 sc0 ls36">x<span class="ff116 fsc ls53 v9">3</span><span class="ff117 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff116 ls1">2<span class="ff115 lsb ws24">x<span class="ff116 fsc v9">4</span></span></span></span></div><div class="t m0 xcb h1c y440 ff115 fs5 fc3 sc0 ls36">x<span class="ff116 fsc ls53 v9">6</span><span class="ff117 ls1d7">=<span class="ff116 fsc lsb v5">1</span></span></div><div class="t m0 xcd h27 y441 ff116 fsc fc3 sc0 lsb">3</div><div class="c x17 y442 wf h8f"><div class="t m0 x0 h90 y443 ff118 fse fc3 sc0 lsb">x</div></div><div class="t m0 xbd h91 y444 ff119 fsd fc3 sc0 ls1d9">6<span class="ff11a fse ls1da ve">=</span><span class="lsb vd">1</span></div><div class="t m0 xa2 h19 y445 ff119 fsd fc3 sc0 lsb">3</div><div class="t m0 xa2 ha y446 ff11b fs5 fc3 sc0 ls36">x<span class="ff11c fsc ls53 v9">1</span><span class="ff11d ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff11c lsb">3<span class="ff11b ws24">x</span><span class="fsc ls175 v9">2</span><span class="ff11d ls4">+</span><span class="ls1">2<span class="ff11b ls69">x</span><span class="fsc ls4d v9">3</span><span class="ff11d ls4">\u2212</span>2<span class="ff11b ls69">x</span></span><span class="fsc v9">5</span></span></span></div><div class="t m0 xa2 ha y447 ff11b fs5 fc3 sc0 ls36">x<span class="ff11c fsc ls53 v9">3</span><span class="ff11d ls6">=<span class="ff11c ls4">1<span class="ff11d">\u2212</span><span class="ls1">2</span></span></span><span class="ls1db">x<span class="ff11c fsc ls1d8 v9">4</span><span class="ff11d ls4">\u2212<span class="ff11c lsb">3</span></span><span class="ls50">x<span class="ff11c fsc lsb v9">6</span></span></span></div><div class="t m0 xa2 h1c y448 ff11b fs5 fc3 sc0 ls36">x<span class="ff11c fsc ls53 v9">6</span><span class="ff11d ls1d7">=<span class="ff11c fsc lsb v5">1</span></span></div><div class="t m0 x28 h27 y449 ff11c fsc fc3 sc0 lsb">3</div><div class="t m0 xce ha y44a ff11e fs5 fc3 sc0 ls36">x<span class="ff11f fsc ls51 v9">1</span><span class="ff120 ls6">+<span class="ff11f lsb">3<span class="ff11e ws24">x</span><span class="fsc ls8f v9">2</span></span>\u2212<span class="ff11f ls1">2</span></span><span class="lsb ws24">x<span class="ff11f fsc ls1c3 v9">3</span><span class="ff120 ls6">+<span class="ff11f ls1">2</span></span>x<span class="ff11f fsc ls1dc v9">5</span><span class="ff120 ls6">=</span><span class="ff11f">0</span></span></div><div class="t m0 xcf h1a y44b ff11e fs5 fc3 sc0 lsb ws24">x<span class="ff11f fsc ls51 v9">3</span><span class="ff120 ls6 v0">+<span class="ff11f ls1">2</span></span><span class="v0">x<span class="ff11f fsc ls1c3 v9">4</span><span class="ff120 ls6">+</span><span class="ff11f">3</span>x<span class="ff11f fsc ls53 v9">6</span><span class="ff120 ls6">=</span><span class="ff11f">1</span></span></div><div class="t m0 x12 h92 y44c ff11e fs5 fc3 sc0 lsb ws24">x<span class="ff11f fsc ls53 v9">6</span><span class="ff120 ls1d7 v0">=</span><span class="ff11f fsc v5">1</span></div><div class="t m0 xd0 h27 y44d ff11f fsc fc3 sc0 lsb">3</div><div class="t m0 xd1 h9 y44e ff121 fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xd1 h9 y44f ff121 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y450 ff121 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y451 ff121 fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y452 ff121 fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xd2 ha y453 ff122 fs5 fc3 sc0 lsbe ws86">13<span class="_15 blank"></span><span class="ff123 lsb">\u2212<span class="ff122 ls1a0">20200</span></span></div><div class="t m0 xd2 ha y454 ff122 fs5 fc3 sc0 lsbc ws18b">0012031</div><div class="t m0 xd2 h93 y455 ff122 fs5 fc3 sc0 ls186 ws1dd">000001 <span class="fsc lsb v5">1</span></div><div class="t m0 xd3 h27 y456 ff122 fsc fc3 sc0 lsb">3</div><div class="t m0 xd2 ha y457 ff122 fs5 fc3 sc0 ls1bc ws18c">0000000</div><div class="t m0 xd4 h9 y458 ff121 fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xd4 h9 y459 ff121 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y45a ff121 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y45b ff121 fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y45c ff121 fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x2 he y45d ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 5<span class="_10 blank"> </span><span class="fs1 fc2 ls9 ws7e">O Exemplo 4 Resolvido por </span></div><div class="t m0 xaa h3 y45e ff1 fs1 fc2 sc0 ls7a">Retro-substituição</div><div class="t m0 x7a ha y45f ff124 fs5 fc3 sc0 lsb ws24">x<span class="ff125 fsc ls53 v9">1</span><span class="ff126 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff125 lsb">3<span class="ff124 ls1d6">r<span class="ff126 ls4">\u2212</span></span><span class="ls2">4<span class="ff124 ls3">s<span class="ff126 ls4">\u2212</span></span><span class="ls1">2<span class="ff124 ls54 ws15e">t, x</span></span></span></span></span></div><div class="t m0 x73 h94 y460 ff125 fsc fc3 sc0 ls55">2<span class="ff126 fs5 ls6 va">=<span class="ff124 ls1e3 ws1dc">r,<span class="_37 blank"> </span>x</span></span><span class="ls53">3<span class="ff126 fs5 ls75 ws25 va">=\u2212<span class="_d blank"></span><span class="ff125 ls1">2<span class="ff124 ls56 ws15f">s, x</span></span></span></span></div><div class="t m0 x31 h95 y460 ff125 fsc fc3 sc0 ls174">4<span class="ff126 fs5 ls6 va">=<span class="ff124 ls56 ws15f">s, x</span></span></div><div class="t m0 xd0 h95 y460 ff125 fsc fc3 sc0 ls51">5<span class="ff126 fs5 ls6 va">=<span class="ff124 ls54 ws15e">t, x</span></span></div><div class="t m0 x83 h96 y460 ff125 fsc fc3 sc0 ls53">6<span class="ff126 fs5 ls1d7 va">=</span><span class="lsb v23">1</span></div><div class="t m0 x4e h27 y461 ff125 fsc fc3 sc0 lsb">3</div><div class="t m0 x30 ha y462 ff127 fs5 fc3 sc0 lsb ws24">x<span class="ff128 fsc ls51 v9">1</span><span class="ff129 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff128 lsb">3<span class="ff127 ws24">x</span><span class="fsc ls175 v9">2</span><span class="ff129 ls4">\u2212</span><span class="ls2">4<span class="ff127 ls176">x</span><span class="fsc ls1d8 v9">4</span><span class="ff129 ls4">\u2212</span><span class="ls1">2<span class="ff127 ls69">x</span></span></span><span class="fsc v9">5</span></span></span></div><div class="t m0 x30 ha y463 ff127 fs5 fc3 sc0 lsb ws24">x<span class="ff128 fsc ls51 v9">3</span><span class="ff129 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff128 ls1">2<span class="ff127 lsb ws24">x<span class="ff128 fsc v9">4</span></span></span></span></div><div class="t m0 x30 h1c y464 ff127 fs5 fc3 sc0 lsb ws24">x<span class="ff128 fsc ls51 v9">6</span><span class="ff129 ls1dd">=</span><span class="ff128 fsc v5">1</span></div><div class="t m0 xb4 h27 y465 ff128 fsc fc3 sc0 lsb">3</div><div class="t m0 xd5 h6b y466 ff12a fs5 fc3 sc0 lsb ws24">x<span class="ff12b fsc ls51 v9">1</span><span class="ff12c ls6 v0">+<span class="ff12b lsb">3<span class="ff12a">x</span><span class="fsc ls1de v9">2</span></span>+<span class="ff12b ls2">4</span></span><span class="v0">x<span class="ff12b fsc ls17f v9">4</span><span class="ff12c ls6">+<span class="ff12b ls1">2</span></span>x<span class="ff12b fsc ls1df v9">5</span><span class="ff12c ls6">=</span><span class="ff12b">0</span></span></div><div class="t m0 xcf h1a y467 ff12a fs5 fc3 sc0 ls1e0">x<span class="ff12b fsc ls53 v9">3</span><span class="ff12c ls6 v0">+<span class="ff12b ls1">2<span class="ff12a lsb ws24">x</span><span class="fsc ls1e1 v9">4</span></span>=<span class="ff12b lsb">0</span></span></div><div class="t m0 x75 h3f y468 ff12a fs5 fc3 sc0 lsb ws24">x<span class="ff12b fsc ls53 v9">6</span><span class="ff12c ls1d7 v0">=</span><span class="ff12b fsc v5">1</span></div><div class="t m0 xd4 h27 y469 ff12b fsc fc3 sc0 lsb">3</div><div class="t m0 xd1 h9 y46a ff12d fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xd1 h9 y46b ff12d fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y46c ff12d fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y46d ff12d fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y46e ff12d fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xd2 ha y46f ff12e fs5 fc3 sc0 lsbd">1304200</div><div class="t m0 xd2 ha y470 ff12e fs5 fc3 sc0 lsc2 wsd4">0012000</div><div class="t m0 xd2 h93 y471 ff12e fs5 fc3 sc0 ls1e2 ws1de">000001 <span class="fsc lsb v5">1</span></div><div class="t m0 xd3 h27 y472 ff12e fsc fc3 sc0 lsb">3</div><div class="t m0 xd2 ha y473 ff12e fs5 fc3 sc0 ls184">0000000</div><div class="t m0 xd4 h9 y474 ff12d fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xd4 h9 y475 ff12d fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y476 ff12d fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y477 ff12d fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y478 ff12d fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xd1 h9 y479 ff12f fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xd1 h9 y47a ff12f fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y47b ff12f fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y47c ff12f fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xd1 h9 y47d ff12f fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xd2 ha y47e ff130 fs5 fc3 sc0 lsbe ws86">13<span class="_15 blank"></span><span class="ff131 lsb">\u2212<span class="ff130 ls1a0">20200</span></span></div><div class="t m0 xd2 ha y47f ff130 fs5 fc3 sc0 lsbc ws18b">0012031</div><div class="t m0 xd2 h93 y480 ff130 fs5 fc3 sc0 ls186 ws1dd">000001 <span class="fsc lsb v5">1</span></div><div class="t m0 xd3 h27 y481 ff130 fsc fc3 sc0 lsb">3</div><div class="t m0 xd2 ha y482 ff130 fs5 fc3 sc0 ls1bc ws18c">0000000</div><div class="t m0 xd4 h9 y483 ff12f fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 xd4 h9 y484 ff12f fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y485 ff12f fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y486 ff12f fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 xd4 h9 y487 ff12f fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x48 h19 y488 ff132 fsd fc3 sc0 lsb">1</div><div class="t m0 x48 h19 y489 ff132 fsd fc3 sc0 lsb">6</div><div class="t m0 x3f h38 ya3 ff4 fs10 fc2 sc0 ls7d ws9a">Capítulo 1 - Sistemas de Equações Linear<span class="_0 blank"></span>es e Matrizes<span class="_12 blank"> </span><span class="ff1 fsf fc1 ls7c ws7f v14">\u2022 \u2022 \u2022<span class="_12 blank"> </span><span class="fs7 lsd v5">35</span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi xb y48a w9 h97" alt="" src="https://files.passeidireto.com/be2469cd-001d-435f-9357-cbee71d47cfb/bga.png"><div class="t m0 xb h4 y48b ff3 fs2 fc2 sc0 lsb ws3c">Substituindo a equação de baixo nas que estão acima, dá</div><div class="t m0 xb h4 y48c ff3 fs2 fc2 sc0 lsb ws1fd">e substituindo a segunda equação na de cima fornece <span class="ff4 ls1e4">x</span>= 1, </div><div class="t m0 xb h4 y48d ff4 fs2 fc2 sc0 ls1e5">y<span class="ff3 lsb ws1fe">= 2, </span><span class="ls1e6">z<span class="ff3 lsb ws1fe">= 3. Isto confere com o resultado obtido pela eliminação</span></span></div><div class="t m0 xb h14 y48e ff3 fs2 fc2 sc0 lsa ws6">de Gauss-Jordan no Exemplo 3 da Seção 1.1.<span class="_38 blank"> </span><span class="ffe fc1 lsb">®</span></div><div class="t m0 xb he y48f ff1 fs7 fc1 sc0 ls71 ws1ff">Sistemas Lineares Homogêneos <span class="ff3 fs2 fc2 lsa ws200">Um sistema de</span></div><div class="t m0 xb h4 y490 ff3 fs2 fc2 sc0 lsb ws1e1">equações lineares é dito <span class="ffd lsa ws1f7">homogêneo <span class="ff3 ws201">se os termos constantes são</span></span></div><div class="t m0 xb h4 y491 ff3 fs2 fc2 sc0 lsa ws6">todos zero; ou seja, o sistema tem a forma</div><div class="t m0 xb h4 y492 ff3 fs2 fc2 sc0 lsb ws202">Cada sistema homogêneo de equações lineares é consistente,</div><div class="t m0 xb h4 y493 ff3 fs2 fc2 sc0 lsa ws203">pois todos sistemas homogêneos têm <span class="ff4 lsb ws17">x<span class="ff3 fs3 ws7 v6">1 <span class="_a blank"> </span></span></span><span class="lsb v0">= 0, <span class="ff4 ls1e7">x</span><span class="fs3 ls1e8 v6">2</span>= 0, ..., <span class="ff4 ls1e9">x<span class="fs3 ls1e8 v6">n</span></span><span class="ws204">= 0</span></span></div><div class="t m0 xb h4 y494 ff3 fs2 fc2 sc0 lsb ws205">como uma solução. Esta solução é chamada <span class="ffd lsa ws206">solução trivial <span class="ff3">ou</span></span></div><div class="t m0 xb h4 y495 ffd fs2 fc2 sc0 lsa ws207">solução nula<span class="ff3 ws208">; se há outras soluções, estas são chamadas </span><span class="lsc">não-</span></div><div class="t m0 xb h4 y496 ffd fs2 fc2 sc0 lsa">triviais<span class="ff3 lsb">.</span></div><div class="t m0 x13 h4 y497 ff3 fs2 fc2 sc0 lsb ws209">Como um sistema linear homogêneo sempre tem a solução</div><div class="t m0 xb h4 y498 ff3 fs2 fc2 sc0 lsa ws6">trivial, só existem duas possibilidades para suas soluções:</div><div class="t m0 x13 h14 y499 ffe fs2 fc1 sc0 ls7e">\u2211<span class="ff3 fc2 lsa ws6">O sistema tem somente a solução trivial.</span></div><div class="t m0 x13 h14 y49a ffe fs2 fc1 sc0 ls7e">\u2211<span class="ff3 fc2 lsa ws20a">O sistema tem infinitas soluções além da solução</span></div><div class="t m0 x37 h4 y49b ff3 fs2 fc2 sc0 lsa">trivial.</div><div class="t m0 xb h4 y49c ff3 fs2 fc2 sc0 lsb ws20b">No caso especial de um sistema linear homogêneo de duas</div><div class="t m0 xb h4 y49d ff3 fs2 fc2 sc0 lsb ws3c">equações em duas incógnitas, digamos</div><div class="t m0 xd6 h98 y49e ff4 fs2 fc2 sc0 ls1ea">a<span class="ff3 fs3 lsb ws7 v6">1 </span><span class="ls2c v0">x<span class="ff3 lsa ws7">+ <span class="_4 blank"> </span></span><span class="lsb ws17">b<span class="ff3 fs3 ws7 v6">1 </span></span>y<span class="ff3 lsb ws6">= 0 <span class="fs10">(<span class="ffd wsd7">a<span class="ff2 fs16 wsd8 v6">1</span><span class="ff2 ws7">, <span class="_4 blank"> </span></span>b<span class="ff2 fs16 ws7 v6">1 <span class="_4 blank"> </span></span><span class="ff2 ls7d ws9a">não ambas nulas</span></span>)</span></span></span></div><div class="t m0 xd6 h4 y49f ff4 fs2 fc2 sc0 lsb ws17">a<span class="ff3 fs3 ws7 v6">2 </span><span class="ls2c">x<span class="ff3 lsa ws7">+ <span class="_4 blank"> </span></span></span>b<span class="ff3 fs3 ws7 v6">2 </span><span class="ls2c">y</span><span class="ff3 ws6">= 0 <span class="fs10">(<span class="ffd wsd7">a<span class="ff2 fs16 wsd8 v6">2</span><span class="ff2 ws7">, <span class="_4 blank"> </span></span>b<span class="ff2 fs16 ws7 v6">2 <span class="_4 blank"> </span></span><span class="ff2 ls7d ws9a">não ambas nulas</span></span>)</span></span></div><div class="t m0 xb h4 y4a0 ff3 fs2 fc2 sc0 lsa ws20c">os gráficos das equações são retas pela origem, e a solução tri-</div><div class="t m0 xb h4 y4a1 ff3 fs2 fc2 sc0 lsb ws3c">vial corresponde ao ponto de corte na origem (Figura 1.2.1).</div><div class="t m0 x8d h2c y4a2 ff3 fs10 fc2 sc0 lsb">(<span class="ff4">a</span><span class="ls7d ws80">) Somente a solução trivial (</span><span class="ff4">b</span><span class="ls7d ws9a">) Infinitas soluções</span></div><div class="t m0 x8d h2d y4a3 ff1 fs2 fc1 sc0 lsb ws9b">Figura 1.2.1</div><div class="t m0 x13 h4 y4a4 ff3 fs2 fc2 sc0 lsb ws20d">Há um caso no qual um sistema homogêneo garantidamente</div><div class="t m0 xb h4 y4a5 ff3 fs2 fc2 sc0 lsa ws20e">tem soluções não-triviais, a saber<span class="_0 blank"></span>, sempre que o sistema envolve</div><div class="t m0 xb h4 y4a6 ff3 fs2 fc2 sc0 lsa ws20f">mais incógnitas que equações. Para ver por que, considere o</div><div class="t m0 xb h4 y4a7 ff3 fs2 fc2 sc0 lsb ws6">seguinte exemplo de quatro equações em cinco incógnitas.</div><div class="t m0 x14 h4 y4a8 ff3 fs2 fc2 sc0 lsb ws210">Resolva o seguinte sistema homogêneo de equações lineares</div><div class="t m0 x14 h4 y4a9 ff3 fs2 fc2 sc0 lsa ws6">usando eliminação de Gauss-Jordan.</div><div class="t m0 xa3 h4 y4aa ff3 fs2 fc2 sc0 ls72">(1)</div><div class="t m0 x14 h35 y4ab ff4 fs2 fc1 sc0 lsb">Solução.</div><div class="t m0 x14 h4 y4ac ff3 fs2 fc2 sc0 lsb ws3c">A<span class="_25 blank"> </span>matriz aumentada para o sistema é</div><div class="t m0 x14 h4 y4ad ff3 fs2 fc2 sc0 lsb ws211">Reduzindo esta matriz à forma escalonada reduzida por linhas,</div><div class="t m0 x14 h4 y4ae ff3 fs2 fc2 sc0 lsa">obtemos</div><div class="t m0 x14 h4 y4af ff3 fs2 fc2 sc0 lsa ws6">O sistema de equações correspondente é</div><div class="t m0 xa3 h4 y4b0 ff3 fs2 fc2 sc0 ls72">(2)</div><div class="t m0 x14 h4 y4b1 ff3 fs2 fc2 sc0 lsa ws6">Resolvendo para as variáveis líderes, obtemos</div><div class="t m0 x14 h4 y4b2 ff3 fs2 fc2 sc0 lsa ws6">Assim, a solução geral é</div><div class="t m0 x14 h14 y4b3 ff3 fs2 fc2 sc0 lsb ws6">Note que a solução trivial é obtida quando <span class="ff4 ls2c">s<span class="ff3">=</span>t</span>= 0.<span class="_39 blank"> </span><span class="ffe fc1">®</span></div><div class="t m0 x15 h4 y4b4 ff3 fs2 fc2 sc0 lsa ws212">O Exemplo 7 ilustra dois aspectos importantes sobre re-</div><div class="t m0 x14 h4 y4b5 ff3 fs2 fc2 sc0 lsba ws213">solução de sistemas homogêneos de equações lineares. O</div><div class="t m0 x14 h4 y4b6 ff3 fs2 fc2 sc0 lsa ws214">primeiro é que nenhuma das operações elementares sobre linhas</div><div class="t m0 x14 h4 y4b7 ff3 fs2 fc2 sc0 lsb ws215">altera a coluna final de zeros da matriz aumentada, de modo que</div><div class="t m0 x14 h4 y4b8 ff3 fs2 fc2 sc0 lsa ws216">o sistema de equações correspondente à forma escalonada</div><div class="t m0 x14 h4 y4b9 ff3 fs2 fc2 sc0 lsb ws217">reduzida por linhas da matriz aumentada também deve ser um</div><div class="t m0 x14 h4 y4ba ff3 fs2 fc2 sc0 lsa ws218">sistema homogêneo [ver sistema (2)]. O segundo é que, depen-</div><div class="t m0 x14 h4 y4bb ff3 fs2 fc2 sc0 lsb ws219">dendo da forma escalonada reduzida por linhas da matriz</div><div class="t m0 x14 h4 y4bc ff3 fs2 fc2 sc0 lsb ws21a">aumentada ter alguma linha nula, o número de equações no sis-</div><div class="t m0 x14 h4 y4bd ff3 fs2 fc2 sc0 lsb ws21b">tema reduzido é menor do que ou igual ao número de equações</div><div class="t m0 x14 h4 y4be ff3 fs2 fc2 sc0 lsa ws21c">do sistema original [compare os sistemas (1) e (2)]. <span class="_0 blank"></span>Assim, se o</div><div class="t m0 x14 h4 y4bf ff3 fs2 fc2 sc0 lsb ws21d">sistema homogêneo dado tiver <span class="ff4 ls1eb">m</span>equações em <span class="ff4 ls1eb">n</span>incógnitas com</div><div class="t m0 x14 h4 y4c0 ff4 fs2 fc2 sc0 lsa ws21e">m < n<span class="ff3 lsb">, e se há </span><span class="ls1ec">r<span class="ff3 lsb">linhas não-nulas na forma escalonada reduzida</span></span></div><div class="t m0 x14 h4 y4c1 ff3 fs2 fc2 sc0 lsb ws21f">por linhas da matriz aumentada, nós teremos <span class="ff4 lsa">r < n<span class="ff3">. Segue-se</span></span></div><div class="t m0 xd7 h99 y4c2 ff133 fs5 fc3 sc0 lsb ws24">x<span class="ff134 fsc ls51 v9">1</span><span class="ff135 ls75 ws25">=\u2212<span class="_d blank"></span><span class="ff133 ls3">s<span class="ff135 ls4">\u2212</span><span class="ls54 ws15e">t, x</span></span></span></div><div class="t m0 xcb h95 y4c3 ff134 fsc fc3 sc0 ls55">2<span class="ff135 fs5 ls6 va">=<span class="ff133 ls56 ws15f">s, x</span></span></div><div class="t m0 xd8 h95 y4c3 ff134 fsc fc3 sc0 ls51">3<span class="ff135 fs5 ls75 ws25 va">=\u2212<span class="_d blank"></span><span class="ff133 ls54 ws15e">t, x</span></span></div><div class="t m0 x25 h94 y4c3 ff134 fsc fc3 sc0 ls6a">4<span class="ff135 fs5 ls6 va">=<span class="ff134 ls1b4">0<span class="ff133 ls79">,x</span></span></span></div><div class="t m0 xd9 h95 y4c3 ff134 fsc fc3 sc0 ls53">5<span class="ff135 fs5 ls6 va">=<span class="ff133 lsb">t</span></span></div><div class="t m0 x42 h9a y4c4 ff136 fs5 fc3 sc0 lsb ws24">x<span class="ff137 fsc ls1ed v9">1</span><span class="ff138 ls75 v0">=\u2212</span><span class="v0">x<span class="ff137 fsc ls55 v9">2</span><span class="ff138 ls6">\u2212</span>x<span class="ff137 fsc v9">5</span></span></div><div class="t m0 x42 h99 y4c5 ff136 fs5 fc3 sc0 lsb ws24">x<span class="ff137 fsc ls1ed v9">3</span><span class="ff138 ls75 ws25">=\u2212</span><span class="ls1ee">x</span><span class="ff137 fsc v9">5</span></div><div class="t m0 x42 ha y4c6 ff136 fs5 fc3 sc0 ls36">x<span class="ff137 fsc ls174 v9">4</span><span class="ff138 ls1ef">=<span class="ff137 lsb">0</span></span></div><div class="t m0 xda ha y4c7 ff139 fs5 fc3 sc0 lsb ws24">x<span class="ff13a fsc ls4d v9">1</span><span class="ff13b ls4">+</span><span class="ls4e">x<span class="ff13a fsc ls179 v9">2</span><span class="ff13b ls6">+</span></span>x<span class="ff13a fsc ls53 v9">5</span><span class="ff13b ls6">=</span><span class="ff13a">0</span></div><div class="t m0 x1e h1a y4c8 ff139 fs5 fc3 sc0 lsb ws24">x<span class="ff13a fsc ls17a v9">3</span><span class="ff13b ls6 v0">+</span><span class="v0">x<span class="ff13a fsc ls53 v9">5</span><span class="ff13b ls6">=</span><span class="ff13a">0</span></span></div><div class="t m0 x26 h1a y4c9 ff139 fs5 fc3 sc0 lsb ws24">x<span class="ff13a fsc ls1f0 v9">4</span><span class="ff13b ls6 v0">=<span class="ff13a lsb">0</span></span></div><div class="t m0 xdb h9 y4ca ff13c fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 xdb h9 y4cb ff13c fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xdb h9 y4cc ff13c fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xdb h9 y4cd ff13c fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 xdb h9 y4ce ff13c fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 x40 ha y4cf ff13d fs5 fc3 sc0 lsbe wsce">110010</div><div class="t m0 x40 ha y4d0 ff13d fs5 fc3 sc0 lsbe ws1f8">001010</div><div class="t m0 x40 ha y4d1 ff13d fs5 fc3 sc0 lsbe ws1f8">000100</div><div class="t m0 x40 ha y4d2 ff13d fs5 fc3 sc0 lsc2 wsc9">000000</div><div class="t m0 x23 h9 y4ca ff13c fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x23 h9 y4cb ff13c fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x23 h9 y4cc ff13c fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x23 h9 y4cd ff13c fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x23 h9 y4ce ff13c fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 x17 h9 y4d3 ff13e fs5 fc3 sc0 lsb">\u23a1</div><div class="t m0 x17 h9 y4d4 ff13e fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y4d5 ff13e fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y4d6 ff13e fs5 fc3 sc0 lsb">\u23a2</div><div class="t m0 x17 h9 y4d7 ff13e fs5 fc3 sc0 lsb">\u23a3</div><div class="t m0 xda ha y4d8 ff13f fs5 fc3 sc0 lsbe wsce">22<span class="_15 blank"></span><span class="ff140 lsb">\u2212<span class="ff13f ls1ff ws1f9">1010</span></span></div><div class="t m0 x56 ha y4d9 ff140 fs5 fc3 sc0 lsb">\u2212<span class="ff13f lsdb">1</span>\u2212<span class="ff13f lsbe wsce">12<span class="_15 blank"></span><span class="ff140 lsb">\u2212<span class="ff13f lsbe">310</span></span></span></div><div class="t m0 xda ha y4da ff13f fs5 fc3 sc0 lsbe wsce">11<span class="_15 blank"></span><span class="ff140 lsb">\u2212<span class="ff13f ls1b2 ws1fa">20<span class="_15 blank"></span><span class="ff140 lsb">\u2212<span class="ff13f lsbe">10</span></span></span></span></div><div class="t m0 xda ha y4db ff13f fs5 fc3 sc0 lsbe ws1fb">001110</div><div class="t m0 x57 h9 y4d3 ff13e fs5 fc3 sc0 lsb">\u23a4</div><div class="t m0 x57 h9 y4d4 ff13e fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x57 h9 y4d5 ff13e fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x57 h9 y4d6 ff13e fs5 fc3 sc0 lsb">\u23a5</div><div class="t m0 x57 h9 y4d7 ff13e fs5 fc3 sc0 lsb">\u23a6</div><div class="t m0 xdb ha y4dc ff141 fs5 fc3 sc0 ls1">2<span class="ff142 lsb ws24">x</span><span class="fsc ls51 v9">1</span><span class="ff143 ls6">+</span>2<span class="ff142 lsb ws24">x</span><span class="fsc ls8f v9">2</span><span class="ff143 ls4a">\u2212<span class="ff142 lsb ws24">x</span></span><span class="fsc ls1f1 v9">3</span><span class="ff143 ls6">+<span class="ff142 lsb ws24">x</span></span><span class="fsc ls51 v9">5</span><span class="ff143 ls6">=</span><span class="lsb">0</span></div><div class="t m0 x17 h1a y4dd ff143 fs5 fc3 sc0 lsb">\u2212<span class="ff142 ws24">x<span class="ff141 fsc ls51 v9">1</span></span><span class="ls4a v0">\u2212<span class="ff142 lsb ws24">x<span class="ff141 fsc ls8f v9">2</span></span><span class="ls6">+<span class="ff141 ls1">2<span class="ff142 lsb ws24">x</span><span class="fsc ls51 v9">3</span></span>\u2212<span class="ff141 lsb">3<span class="ff142 ws24">x</span><span class="fsc ls6a v9">4</span></span>+<span class="ff142 lsb ws24">x<span class="ff141 fsc ls51 v9">5</span></span>=<span class="ff141 lsb">0</span></span></span></div><div class="t m0 x56 h1a y4de ff142 fs5 fc3 sc0 lsb ws24">x<span class="ff141 fsc ls51 v9">1</span><span class="ff143 ls4a v0">+</span><span class="v0">x<span class="ff141 fsc ls8f v9">2</span><span class="ff143 ls6">\u2212<span class="ff141 ls1">2</span></span>x<span class="ff141 fsc ls1f1 v9">3</span><span class="ff143 ls6">\u2212</span>x<span class="ff141 fsc ls51 v9">5</span><span class="ff143 ls6">=</span><span class="ff141">0</span></span></div><div class="t m0 xb5 h1a y4df ff142 fs5 fc3 sc0 lsb ws24">x<span class="ff141 fsc ls51 v9">3</span><span class="ff143 lsd5 v0">+</span><span class="v0">x<span class="ff141 fsc ls6a v9">4</span><span class="ff143 ls6">+</span>x<span class="ff141 fsc ls51 v9">5</span><span class="ff143 ls6">=</span><span class="ff141">0</span></span></div><div class="t m0 x3f he y4e0 ff1 fs7 fc1 sc0 ls71 ws7d">EXEMPLO 7<span class="_34 blank"> </span><span class="fs1 fc2 ls7a ws7e">Eliminação de Gauss-Jordan</span></div><div class="t m0 x8c h60 y4e1 ff144 fs11 fc3 sc0 lsb">y</div><div class="t m0 xdc h60 y4e2 ff144 fs11 fc3 sc0 lsb">x</div><div class="t m0 xdd h9b y4e3 ff56 fs11 fc3 sc0 lsb">e</div><div class="t m0 xc3 h62 y4e4 ff144 fs11 fc3 sc0 lsb ws1fc">a<span class="ff145 fs19 ls1f2 v8">1</span><span class="ws7">x <span class="ff145">+</span> b<span class="ff145 fs19 ls154 v8">1</span>y <span class="ff145">= 0</span></span></div><div class="t m0 xde h9c y4e5 ff144 fsa fc3 sc0 lsb ws22">a<span class="ff145 fs19 ls154 v8">2</span><span class="fs11 ws7">x <span class="ff145">+</span> b<span class="ff145 fs19 ls1f3 v8">2</span>y <span class="ff145">= 0</span></span></div><div class="t m0 xd2 h60 y4e6 ff146 fs11 fc3 sc0 lsb">y</div><div class="t m0 x46 h60 y4e7 ff146 fs11 fc3 sc0 lsb">x</div><div class="t m0 xd6 h62 y4e8 ff146 fs11 fc3 sc0 lsb ws1fc">a<span class="ff147 fs19 wsf3 v8">1</span><span class="ws7">x <span class="ff147">+</span> b<span class="ff147 fs19 ls154 v8">1</span>y <span class="ff147">= 0</span></span></div><div class="t m0 xd6 h9d y4e9 ff146 fs11 fc3 sc0 ls27">a<span class="ff147 fs19 ls1f4 v8">2</span><span class="lsb ws7 v0">x <span class="ff147">+</span> b<span class="ff147 fs19 ls1f5 v8">2</span>y <span class="ff147">= 0</span></span></div><div class="t m0 x4a h9e y4ea ff148 fs5 fc3 sc0 lsb ws24">a<span class="ff149 fsc ws8d v9">11 </span><span class="v0">x<span class="ff149 fsc ls1f6 v9">1</span><span class="ff14a ls9f">+</span>a<span class="ff149 fsc ws90 v9">12 </span><span class="ls36">x<span class="ff149 fsc ls1f7 v9">2</span><span class="ff14a ls91 ws91">+···+ </span></span>a<span class="ff149 fsc v9">1<span class="ff148 ls6c">n</span></span>x<span class="fsc ls1f8 v9">n</span><span class="ff14a ls6">=</span><span class="ff149">0</span></span></div><div class="t m0 x4a h1a y4eb ff148 fs5 fc3 sc0 ls1f9">a<span class="ff149 fsc lsb ws8d v9">21 </span><span class="lsb ws24 v0">x<span class="ff149 fsc ls1f6 v9">1</span><span class="ff14a ls9f">+</span>a<span class="ff149 fsc ws90 v9">22 </span><span class="ls36">x<span class="ff149 fsc ls1f7 v9">2</span><span class="ff14a ls91 ws92">+···+ </span></span>a<span class="ff149 fsc ls94 v9">2<span class="ff148 ls1fa">n</span></span>x<span class="fsc lsa6 v9">n</span><span class="ff14a ls6">=</span><span class="ff149">0</span></span></div><div class="t m0 x38 h16 y4ec ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 x38 h16 y4ed ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 x38 h32 y4ee ff148 fs5 fc3 sc0 lsa9">.<span class="lsb v10">.</span></div><div class="t m0 x4b h16 y4ed ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 x4b h32 y4ee ff148 fs5 fc3 sc0 lsaa">.<span class="lsb v10">.</span></div><div class="t m0 x31 h16 y4ed ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 x31 h32 y4ee ff148 fs5 fc3 sc0 ls1fb">.<span class="lsb v10">.</span></div><div class="t m0 xd3 h16 y4ed ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 xd3 h16 y4ee ff148 fs5 fc3 sc0 lsb">.</div><div class="t m0 x4a h9f y4ef ff148 fs5 fc3 sc0 lsb ws24">a<span class="fsc v9">m<span class="ff149 lsac">1</span></span><span class="v0">x<span class="ff149 fsc ls51 v9">1</span><span class="ff14a lsad">+</span>a<span class="fsc v9">m<span class="ff149 lsae">2</span></span><span class="ls36">x<span class="ff149 fsc ls55 v9">2</span><span class="ff14a ls91 ws93">+···+ </span><span class="lsaf">a</span></span><span class="fsc ws94 v9">mn </span><span class="ls36">x<span class="fsc ls1fc v9">n</span><span class="ff14a ls6">=</span></span><span class="ff149">0</span></span></div><div class="t m0 x7b ha y4f0 ff14b fs5 fc3 sc0 ls5">x<span class="ff14c ls6">=<span class="ff14d ls4">9<span class="ff14c">\u2212</span></span></span><span class="ls3">y<span class="ff14c ls4">\u2212<span class="ff14d ls1">2</span></span><span class="lsb">z</span></span></div><div class="t m0 x7b h16 y4f1 ff14b fs5 fc3 sc0 ls5">y<span class="ff14c ls75">=\u2212</span></div><div class="t m0 xc4 h27 y4f2 ff14d fsc fc3 sc0 lsb">17</div><div class="t m0 xdf h1d y4f3 ff14d fsc fc3 sc0 ls1fd">2<span class="ff14c fs5 ls1fe v5">+</span><span class="lsb vb">7</span></div><div class="t m0 x47 h1e y4f3 ff14d fsc fc3 sc0 ls59">2<span class="ff14b fs5 lsb v5">z</span></div><div class="t m0 x36 ha y4f4 ff14b fs5 fc3 sc0 ls8c">z<span class="ff14c ls6">=<span class="ff14d lsb">3</span></span></div><div class="t m0 x44 ha y4f5 ff14e fs5 fc3 sc0 ls5">x<span class="ff14f ls6">+</span>y<span class="ff14f lscc">+<span class="ff150 ls1">2</span></span><span class="ls8c">z<span class="ff14f lsc8">=<span class="ff150 lsb">9</span></span></span></div><div class="t m0 xc0 ha0 y4f6 ff14e fs5 fc3 sc0 ls5">y<span class="ff14f lscd">\u2212<span class="ff150 fsc lsb v5">7</span></span></div><div class="t m0 xdf h1e y4f7 ff150 fsc fc3 sc0 lsce">2<span class="ff14e fs5 ls8c v5">z<span class="ff14f ls75">=\u2212</span></span></div><div class="t m0 xca h27 y4f8 ff150 fsc fc3 sc0 lsb">17</div><div class="t m0 xca h27 y4f7 ff150 fsc fc3 sc0 lsb">2</div><div class="t m0 x50 ha y4f9 ff14e fs5 fc3 sc0 ls8c">z<span class="ff14f lsc8">=<span class="ff150 lsb">3</span></span></div><div class="t m0 xb h2a ya3 ff1 fs7 fc1 sc0 lsd ws2e">36 <span class="fsf ls7c ws7f vf">\u2022 \u2022 \u2022<span class="_12 blank"> </span></span><span class="ff3 fs10 fc2 ls7d ws80">Álgebra Linear com <span class="_0 blank"></span>Aplicações</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-41.952800,-41.952800]}'></div></div>
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