<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/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg1.png"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls18 ws0">EA<span class="_0 blank"></span>E0<span class="_0 blank"></span>207<span class="_0 blank"></span>:<span class="_1 blank"> </span>M<span class="_0 blank"></span>ate<span class="_0 blank"></span>mát<span class="_0 blank"></span>ica A<span class="_0 blank"></span>pli<span class="_0 blank"></span>cad<span class="_0 blank"></span>a à Eco<span class="_0 blank"></span>no<span class="_0 blank"></span>mia</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls18 ws1">Aula 2:<span class="_2 blank"> </span>Sistemas de Equações Linea<span class="_0 blank"></span>res</div><div class="t m0 x3 h4 y4 ff1 fs1 fc1 sc0 ls18 ws1">Ma<span class="_0 blank"></span>rcos Y. Nakaguma</div><div class="t m0 x4 h4 y5 ff1 fs1 fc1 sc0 ls18 ws2">04/08/2017</div><div class="t m0 x5 h5 y6 ff1 fs2 fc1 sc0 ls18">1</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h6 y8 ff1 fs3 fc1 sc0 ls18 ws3">S<span class="_0 blank"></span>i<span class="_0 blank"></span>s<span class="_0 blank"></span>te<span class="_0 blank"></span>m<span class="_3 blank"></span>a<span class="_0 blank"></span>s d<span class="_0 blank"></span>e E<span class="_3 blank"></span>q<span class="_0 blank"></span>u<span class="_0 blank"></span>a<span class="_0 blank"></span>ç<span class="_0 blank"></span>õ<span class="_0 blank"></span>e<span class="_0 blank"></span>s L<span class="_0 blank"></span>i<span class="_0 blank"></span>n<span class="_0 blank"></span>e<span class="_0 blank"></span>a<span class="_3 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>s</div><div class="t m0 x5 h5 y9 ff1 fs2 fc1 sc0 ls18">2</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 yc ff1 fs1 fc2 sc0 ls18 ws5">Sistemas linea<span class="_0 blank"></span>res <span class="fc1 ws6">são<span class="_4 blank"> </span>sistemas<span class="_4 blank"> </span>fo<span class="_0 blank"></span>rmados<span class="_4 blank"> </span>por<span class="_5 blank"> </span><span class="fc2 ws7">equações linea<span class="_0 blank"></span>res <span class="fc1">da</span></span></span></div><div class="t m0 x8 h4 yd ff1 fs1 fc1 sc0 ls18 ws8">seguinte fo<span class="_0 blank"></span>rma:</div><div class="t m0 x9 h7 ye ff2 fs1 fc1 sc0 ls0">a<span class="ff1 fs2 ls1 v1">1</span><span class="ls2">x<span class="ff1 fs2 ls3 v1">1</span><span class="ff3 fs4 ls4 v0">+</span></span><span class="v0">a<span class="ff1 fs2 ls1 v1">2</span><span class="ls18 ws2">x<span class="ff1 fs2 ls5 v1">2</span><span class="ff3 fs4 ls6">+</span><span class="ff4 ws9">... <span class="ff3 fs4 ls7">+</span></span><span class="ls8">a<span class="fs2 ls9 v1">n</span></span>x<span class="fs2 lsa v1">n</span><span class="ff3 fs4 lsb">=</span><span class="lsc">b</span><span class="ff4">,</span></span></span></div><div class="t m0 x8 h4 yf ff1 fs1 fc1 sc0 ls18 ws2">onde:</div><div class="t m0 xa h8 y10 ff5 fs5 fc0 sc0 lsd">I<span class="ff2 fs6 fc1 lse v1">a<span class="ff1 fs2 lsf v1">1</span><span class="ff4 ls18 wsa">, ..., <span class="ff2 ls10">a<span class="fs2 ls11 v1">n</span></span><span class="ff1 wsb">são <span class="fc2 wsc">pa<span class="_0 blank"></span>râmetros <span class="fc1 wsd">(constantes);</span></span></span></span></span></div><div class="t m0 xa h8 y11 ff5 fs5 fc0 sc0 lsd">I<span class="ff2 fs6 fc1 ls12 v1">x<span class="ff1 fs2 lsf v1">1</span><span class="ff4 ls18 wsa">, ..., </span>x<span class="fs2 ls11 v1">n</span><span class="ff1 ls18 wse">são <span class="fc2 wsf">váriáveis </span><span class="wsd">(incógnitas).</span></span></span></div><div class="t m0 x8 h4 y12 ff1 fs1 fc2 sc0 ls18 ws2">Exemplos:</div><div class="t m0 xb h4 y13 ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2 ls2">x</span><span class="fs2 ls13 v1">1</span><span class="ff6 fs4 ls4">\ue000</span>3<span class="ff2 ls2">x</span><span class="fs2 ls14 v1">2</span><span class="ff3 fs4 lsb">=</span>8</div><div class="t m0 x8 h4 y14 ff1 fs1 fc1 sc0 ls18">e</div><div class="t m0 xc h4 y15 ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2 ls2">x</span><span class="fs2 ls15 v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="v0">7<span class="ff2">x</span><span class="fs2 ls16 v1">2</span><span class="ff6 fs4 ls4">\ue000</span>3<span class="ff2">x</span><span class="fs2 ls17 v1">3</span><span class="ff3 fs4 lsb">=</span>5</span></div><div class="t m0 x5 h5 y16 ff1 fs2 fc1 sc0 ls18">3</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg2.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 y18 ff1 fs1 fc1 sc0 ls18 ws1">Em um sistema de equações linea<span class="_0 blank"></span>res, to<span class="_6 blank"> </span>das as va<span class="_0 blank"></span>riáveis <span class="ff2 ls2">x</span><span class="fs2 ls19 v1">1</span>, ..., <span class="ff2 ls2">x<span class="fs2 ls18 v1">n</span></span></div><div class="t m0 x8 h9 y19 ff1 fs1 fc1 sc0 ls18 ws11">devem entra<span class="_0 blank"></span>r <span class="fc2 ws12 v0">linearmente <span class="fc1 ws13">nas equações.</span></span></div><div class="t m0 x8 h4 y1a ff1 fs1 fc1 sc0 ls18 ws13">P<span class="_0 blank"></span>or exemplo, as va<span class="_0 blank"></span>riáveis não p<span class="_6 blank"> </span>odem ser elevadas à nenhuma</div><div class="t m0 x8 h4 y1b ff1 fs1 fc1 sc0 ls18 ws13">p<span class="_6 blank"> </span>otência nem serem multiplicadas por outras va<span class="_0 blank"></span>riáveis.</div><div class="t m0 x8 h4 y1c ff1 fs1 fc1 sc0 ls18 ws13">As seguintes equações são <span class="fc2 ws2">não-linea<span class="_0 blank"></span>res<span class="fc1">:</span></span></div><div class="t m0 xd ha y1d ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2 ls1a">x</span><span class="fs2 v2">2</span></div><div class="t m0 x4 hb y1e ff1 fs2 fc1 sc0 ls1b">1<span class="ff6 fs4 ls4 v3">\ue000</span><span class="fs1 ls18 ws2 v3">4<span class="ff2 ls2">x</span></span><span class="ls19 v4">1</span><span class="ff2 fs1 ls2 v3">x</span><span class="ls1c v4">2</span><span class="ff3 fs4 ls4 v3">+<span class="ff2 fs1 ls1a">x</span></span><span class="ls18 v5">2</span></div><div class="t m0 xe hc y1f ff1 fs2 fc1 sc0 ls1d">2<span class="ff3 fs4 lsb v3">=</span><span class="fs1 ls18 v3">0</span></div><div class="t m0 x8 h4 y20 ff1 fs1 fc1 sc0 ls18">e</div><div class="t m0 xf hd y21 ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls1e v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="v0">x<span class="ff1 fs2 ls1f v1">1</span><span class="ff1 ls18 ws14">ln <span class="ff3 fs4 ls20 v0">(</span></span><span class="ls18 ws2">x<span class="ff1 fs2 ls21 v1">2</span><span class="ff3 fs4 ls22 v0">)<span class="ls23 v0">=</span></span><span class="ff1">3</span></span></span></div><div class="t m0 x5 h5 y6 ff1 fs2 fc1 sc0 ls18">4</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y22 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 y23 ff1 fs1 fc1 sc0 ls18 ws15">Os <span class="fc2 ws16">sistemas linea<span class="_0 blank"></span>res <span class="fc1 ws17">são bastante estudados p<span class="_6 blank"> </span>ois p<span class="_6 blank"> </span>odem ser resolvidos</span></span></div><div class="t m0 x8 h4 y24 ff1 fs1 fc1 sc0 ls18 ws18">de<span class="_5 blank"> </span>forma<span class="_5 blank"> </span>exata,<span class="_4 blank"> </span>i.e.<span class="_2 blank"> </span>p ossuem<span class="_5 blank"> </span>solução<span class="_4 blank"> </span>em<span class="_4 blank"> </span>fo<span class="_0 blank"></span>rma<span class="_5 blank"> </span>fechada:</div><div class="t m0 x10 he y25 ff7 fs1 fc1 sc0 ls24">\ue01a<span class="ff2 ls18 ws2 v6">ax</span><span class="ff1 fs2 ls25 v7">1</span><span class="ff3 fs4 ls4 v6">+</span><span class="ff2 ls18 ws2 v6">b<span class="_0 blank"></span>x<span class="ff1 fs2 ls26 v1">2</span><span class="ff3 fs4 ls27">=</span>e</span></div><div class="t m0 x11 hf y26 ff2 fs1 fc1 sc0 ls18 ws2">cx<span class="ff1 fs2 ls28 v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="v0">dx<span class="ff1 fs2 ls29 v1">2</span><span class="ff3 fs4 ls27">=</span><span class="ls2a">f<span class="ff6 fs4 ls2b v8">)</span><span class="ls2 v8">x</span><span class="ff1 fs2 ls2c v9">1</span><span class="ff3 fs4 ls2d v8">=</span></span><span class="ws19 va">de <span class="ff6 fs4 ls4">\ue000</span>bf</span></span></div><div class="t m0 x12 h10 y27 ff2 fs1 fc1 sc0 ls18 ws1a">ad <span class="ff6 fs4 ls4">\ue000</span><span class="ws1b">cb <span class="ff1 ls2e vb">e</span><span class="ls2 vb">x</span><span class="ff1 fs2 ls2f vc">2</span><span class="ff3 fs4 ls30 vb">=</span><span class="ws1c vd">af <span class="ff6 fs4 ls4">\ue000</span>ce</span></span></div><div class="t m0 x13 h11 y27 ff2 fs1 fc1 sc0 ls18 ws1d">ad <span class="ff6 fs4 ls4">\ue000</span>cb</div><div class="t m0 x8 h4 y28 ff1 fs1 fc1 sc0 ls18 ws1e">P<span class="_0 blank"></span>or outro lado, observe que nem semp<span class="_0 blank"></span>re é p<span class="_6 blank"> </span>ossível deriva<span class="_0 blank"></span>r</div><div class="t m0 x8 h4 y29 ff1 fs1 fc1 sc0 ls18 ws1f">analiticamente a solução de um <span class="fc2 v0">sistema não-linea<span class="_0 blank"></span>r<span class="fc1">:</span></span></div><div class="t m0 x14 he y2a ff7 fs1 fc1 sc0 ls31">\ue01a<span class="ff2 ls18 ws20 v6">ax <span class="ff1 fs2 ve">3</span></span></div><div class="t m0 x15 h12 y2b ff1 fs2 fc1 sc0 ls32">1<span class="ff3 fs4 ls4 vf">+<span class="ff2 fs1 ls18 ws21">b<span class="_0 blank"></span>x <span class="ff1 fs2 ve">2</span></span></span></div><div class="t m0 x16 h13 y2c ff1 fs2 fc1 sc0 ls33">2<span class="ff3 fs4 lsb vf">=<span class="ff2 fs1 ls18">e</span></span></div><div class="t m0 x17 h14 y2d ff2 fs1 fc1 sc0 ls34">c<span class="ff1 ls18 ws22">ln <span class="ff3 fs4 ls35 v0">(</span></span><span class="ls36 v0">x<span class="ff1 fs2 ls37 v1">1</span><span class="ff3 fs4 ls38 v0">)<span class="ls39 v0">+</span></span><span class="ls18 ws2">dx<span class="ff1 fs2 ls3a v1">2</span><span class="ff3 fs4 ls27">=</span><span class="ls2a">f<span class="ff6 fs4 ls3b v8">)</span></span></span><span class="ff8 ls3c ws23 v8">???<span class="_0 blank"></span></span></span></div><div class="t m0 x8 h4 y2e ff1 fs1 fc1 sc0 ls18 ws13">O comp<span class="_6 blank"> </span>o<span class="_0 blank"></span>rtamento de sistemas não-linea<span class="_0 blank"></span>res é, em geral, estudado</div><div class="t m0 x8 h4 y2f ff1 fs1 fc1 sc0 ls18 ws1">numericamente ou através de uma ap<span class="_0 blank"></span>roximação linea<span class="_0 blank"></span>r.</div><div class="t m0 x5 h5 y30 ff1 fs2 fc1 sc0 ls18">5</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 y31 ff1 fs1 fc1 sc0 ls18 ws24">Um sistema linea<span class="_0 blank"></span>r de <span class="ff2 fc2 ls3d">m</span><span class="fc2 ws25">equações </span><span class="ls3e">e<span class="ff2 fc2 ls3f">n</span></span><span class="fc2 ws26">incógnitas </span><span class="ws27">p o de<span class="_4 blank"> </span>ser<span class="_5 blank"> </span>escrito<span class="_4 blank"> </span>de</span></div><div class="t m0 x8 h4 y32 ff1 fs1 fc1 sc0 ls18 ws1">fo<span class="_0 blank"></span>rma geral como:</div><div class="t m0 x18 he y33 ff7 fs1 fc1 sc0 ls18">8</div><div class="t m0 x18 he y34 ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y35 ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y36 ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y37 ff7 fs1 fc1 sc0 ls18"><</div><div class="t m0 x18 he y38 ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y39 ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y3a ff7 fs1 fc1 sc0 ls18">></div><div class="t m0 x18 he y3b ff7 fs1 fc1 sc0 ls18">:</div><div class="t m0 x19 h15 y3c ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="ls2">x<span class="ff1 fs2 ls41 v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="ls0 v0">a<span class="ff1 fs2 ls18 ws28 v1">12<span class="_7 blank"> </span></span><span class="ls2">x<span class="ff1 fs2 ls42 v1">2</span><span class="ff3 fs4 ls4">+</span><span class="ff4 ls18 ws29">... <span class="ff3 fs4 ls7">+</span></span></span>a<span class="ff1 fs2 ls43 v1">1<span class="ff2 ls44">n</span></span><span class="ls2">x<span class="fs2 ls45 v1">n</span><span class="ff3 fs4 lsb">=</span><span class="ls18 ws2">b<span class="ff1 fs2 v1">1</span></span></span></span></span></div><div class="t m0 x19 h16 y3d ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_7 blank"> </span></span><span class="ls2">x<span class="ff1 fs2 ls3 v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="ls0 v0">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_7 blank"> </span></span><span class="ls2">x<span class="ff1 fs2 ls42 v1">2</span><span class="ff3 fs4 ls4">+</span><span class="ff4 ls18 ws29">... <span class="ff3 fs4 ls7">+</span></span></span>a<span class="ff1 fs2 ls43 v1">2<span class="ff2 ls44">n</span></span><span class="ls2">x<span class="fs2 ls45 v1">n</span><span class="ff3 fs4 lsb">=</span><span class="ls18 ws2">b<span class="ff1 fs2 v1">2</span></span></span></span></span></div><div class="t m0 x1a h4 y3e ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1a h4 y3f ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1a h4 y40 ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1b h17 y41 ff2 fs1 fc1 sc0 ls0">a<span class="fs2 ls46 v1">m<span class="ff1 ls19">1</span></span><span class="ls18 ws2">x<span class="ff1 fs2 ls25 v1">1</span><span class="ff3 fs4 ls6 v0">+</span></span><span class="v0">a<span class="fs2 ls46 v1">m<span class="ff1 lsf">2</span></span><span class="ls2">x<span class="ff1 fs2 ls47 v1">2</span><span class="ff3 fs4 ls4">+</span><span class="ff4 ls18 ws2a">... <span class="ff3 fs4 ls6">+</span></span></span>a<span class="fs2 ls18 ws2b v1">m n<span class="_7 blank"> </span></span><span class="ls2">x<span class="fs2 ls48 v1">n</span><span class="ff3 fs4 lsb">=</span><span class="ls18 ws2">b<span class="fs2 v1">m</span></span></span></span></div><div class="t m0 x8 h4 y42 ff1 fs1 fc1 sc0 ls18 ws24">ou em fo<span class="_0 blank"></span>rma matricial:</div><div class="t m0 x1c he y43 ff7 fs1 fc1 sc0 ls18">2</div><div class="t m0 x1c he y44 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x1c he y45 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x1c he y46 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x1c he y47 ff7 fs1 fc1 sc0 ls18">4</div><div class="t m0 x1d h18 y48 ff2 fs1 fc1 sc0 ls8">a<span class="ff1 fs2 ls18 ws2c v1">11<span class="_8 blank"> </span></span><span class="ls49">a<span class="ff1 fs2 ls18 ws2c v1">12<span class="_9 blank"> </span></span><span class="ff4 ls18 ws2d">... </span><span class="ls0">a<span class="ff1 fs2 ls43 v1">1<span class="ff2 ls18">n</span></span></span></span></div><div class="t m0 x1d h18 y49 ff2 fs1 fc1 sc0 ls8">a<span class="ff1 fs2 ls18 ws2c v1">21<span class="_8 blank"> </span></span><span class="ls49">a<span class="ff1 fs2 ls18 ws2c v1">22<span class="_a blank"> </span></span><span class="ls0">a<span class="ff1 fs2 ls4a v1">2<span class="ff2 ls18">n</span></span></span></span></div><div class="t m0 x1e h4 y4a ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1e h4 y4b ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1e h4 y4c ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x1f h18 y4d ff2 fs1 fc1 sc0 ls0">a<span class="fs2 ls46 v1">m<span class="ff1 ls4b">1</span></span><span class="ls40">a<span class="fs2 ls4c v1">m<span class="ff1 ls4d">2</span></span><span class="ls49">a<span class="fs2 ls18 ws2e v1">m n</span></span></span></div><div class="t m0 x20 he y4e ff7 fs1 fc1 sc0 ls18">3</div><div class="t m0 x20 he y4f ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x20 he y50 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x20 he y51 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x20 he y52 ff7 fs1 fc1 sc0 ls18">5</div><div class="t m0 x1c he y53 ff7 fs1 fc1 sc0 ls4e">|<span class="ls18 ws2f v0">{<span class="_6 blank"></span>z }</span></div><div class="t m0 x21 h5 y54 ff1 fs2 fc1 sc0 ls18 ws30">m a t r iz<span class="_5 blank"> </span>de<span class="_5 blank"> </span>co e \u2026<span class="_6 blank"> </span>ci e n t e s<span class="_b blank"> </span>(<span class="ff2 ls4f">A</span>)</div><div class="t m0 x22 he y55 ff7 fs1 fc1 sc0 ls18">2</div><div class="t m0 x22 he y56 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x22 he y57 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x22 he y58 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x22 he y59 ff7 fs1 fc1 sc0 ls18">4</div><div class="t m0 x23 h18 y5a ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls18 v1">1</span></div><div class="t m0 x23 h18 y5b ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls18 v1">2</span></div><div class="t m0 x24 h4 y5c ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x24 h4 y5d ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x24 h4 y5e ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x23 h18 y5f ff2 fs1 fc1 sc0 ls2">x<span class="fs2 ls18 v1">n</span></div><div class="t m0 x25 he y60 ff7 fs1 fc1 sc0 ls18">3</div><div class="t m0 x25 he y61 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x25 he y62 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x25 he y63 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x25 h19 y64 ff7 fs1 fc1 sc0 ls50">5<span class="ff3 fs4 ls51 vb">=</span><span class="ls18 v10">2</span></div><div class="t m0 x26 he y65 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x26 he y66 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x26 he y67 ff7 fs1 fc1 sc0 ls18">6</div><div class="t m0 x26 he y68 ff7 fs1 fc1 sc0 ls18">4</div><div class="t m0 x27 h18 y69 ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x27 h18 y6a ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 v1">2</span></div><div class="t m0 x28 h4 y6b ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x28 h4 y6c ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x28 h4 y6d ff1 fs1 fc1 sc0 ls18">.</div><div class="t m0 x29 h18 y6e ff2 fs1 fc1 sc0 ls18 ws2">b<span class="fs2 v1">m</span></div><div class="t m0 x2a he y6f ff7 fs1 fc1 sc0 ls18">3</div><div class="t m0 x2a he y70 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x2a he y71 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x2a he y72 ff7 fs1 fc1 sc0 ls18">7</div><div class="t m0 x2a h1a y73 ff7 fs1 fc1 sc0 ls52">5<span class="ff6 fs4 ls53 vb">$</span><span class="ff2 ls18 ws2 vb">A<span class="ff9 ls54">x<span class="ff3 fs4 ls27">=</span><span class="ls18">b</span></span></span></div><div class="t m0 x5 h5 y74 ff1 fs2 fc1 sc0 ls18">6</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg3.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 y75 ff1 fs1 fc1 sc0 ls18 ws31">De fo<span class="_0 blank"></span>rma geral, estamos interessados nas seguintes questões:</div><div class="t m0 x2b h1b y76 ff2 fs6 fc0 sc0 ls55">i<span class="ff4 ls56">.<span class="ff1 fc1 ls18 ws32">Quando um sistema de equações linea<span class="_0 blank"></span>res tem <span class="fc2 wsd">solução</span><span class="ws33">? Quantas</span></span></span></div><div class="t m0 x2c h1b y77 ff1 fs6 fc1 sc0 ls18 ws32">soluções existem?</div><div class="t m0 x2d h1b y78 ff2 fs6 fc0 sc0 ls18 ws34">ii <span class="ff4 ls57">.</span><span class="ff1 fc1 ws32">Quais condições ga<span class="_0 blank"></span>rantem a existência de <span class="fc2 ws35">p elo<span class="_5 blank"> </span>menos<span class="_5 blank"> </span>uma<span class="_5 blank"> </span></span><span class="wsd">solução?</span></span></div><div class="t m0 x2e h1b y79 ff2 fs6 fc0 sc0 ls18 ws36">iii <span class="ff4 ls58">.</span><span class="ff1 fc1 ws32">Quais condições ga<span class="_0 blank"></span>rantem a existência de uma <span class="fc2 ws37">única </span><span class="wsd">solução?</span></span></div><div class="t m0 x5 h5 y7a ff1 fs2 fc1 sc0 ls18">7</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y22 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res</div><div class="t m0 x8 h4 y7b ff1 fs1 fc1 sc0 ls18 ws13">Existem três méto<span class="_6 blank"> </span>dos básicos que po<span class="_6 blank"> </span>dem ser utilizados pa<span class="_0 blank"></span>ra resolver</div><div class="t m0 x8 h4 y7c ff1 fs1 fc1 sc0 ls18 ws31">sistemas linea<span class="_0 blank"></span>res:</div><div class="t m0 x2d h1b y7d ff1 fs6 fc0 sc0 ls18 wsd">1<span class="ff4 ls59">.</span><span class="fc1">Substituição;</span></div><div class="t m0 x2d h1b y7e ff1 fs6 fc0 sc0 ls18 wsd">2<span class="ff4 ls59">.</span><span class="fc1 ws32">Eliminação de V<span class="_0 blank"></span>ariáveis (Eliminação de Gauss-Jo<span class="_0 blank"></span>rdan); e</span></div><div class="t m0 x2d h1b y7f ff1 fs6 fc0 sc0 ls18 wsd">3<span class="ff4 ls59">.</span><span class="fc1 ws38">Méto dos<span class="_b blank"> </span>Matriciais<span class="_5 blank"> </span>(Solução<span class="_5 blank"> </span>p or<span class="_b blank"> </span>Escalonamento).</span></div><div class="t m0 x5 h5 y9 ff1 fs2 fc1 sc0 ls18">8</div></div><div class="c x0 ya w2 h2"><div class="t m0 x1e h6 y80 ff1 fs3 fc1 sc0 ls18 ws3">S<span class="_0 blank"></span>i<span class="_0 blank"></span>s<span class="_0 blank"></span>te<span class="_0 blank"></span>m<span class="_3 blank"></span>a<span class="_0 blank"></span>s d<span class="_0 blank"></span>e E<span class="_3 blank"></span>q<span class="_0 blank"></span>u<span class="_0 blank"></span>a<span class="_0 blank"></span>ç<span class="_0 blank"></span>õ<span class="_0 blank"></span>e<span class="_0 blank"></span>s L<span class="_0 blank"></span>i<span class="_0 blank"></span>n<span class="_0 blank"></span>e<span class="_0 blank"></span>a<span class="_3 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>s:</div><div class="t m0 xf h6 y81 ff1 fs3 fc1 sc0 ls18 ws39">E<span class="_0 blank"></span>x<span class="_0 blank"></span>e<span class="_0 blank"></span>m<span class="_3 blank"></span>p<span class="_0 blank"></span>l<span class="_0 blank"></span>o<span class="_0 blank"></span>s</div><div class="t m0 x5 h5 y16 ff1 fs2 fc1 sc0 ls18">9</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg4.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws0">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as co<span class="_0 blank"></span>m um<span class="_0 blank"></span>a Ú<span class="_0 blank"></span>nica S<span class="_0 blank"></span>olu<span class="_0 blank"></span>ção</div><div class="t m0 x8 h4 y82 ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 1<span class="fc1 ls5a">:<span class="ff7 ls5b v11">\ue01a</span><span class="ff2 ls2 v12">x</span><span class="fs2 ls5c v13">1</span><span class="ff6 fs4 ls4 v12">\ue000</span><span class="ls18 ws2 v12">2<span class="ff2 ls2">x</span></span><span class="fs2 ls14 v13">2</span><span class="ff3 fs4 lsb v12">=</span><span class="ls18 v12">8</span></span></div><div class="t m0 x0 h1c y83 ff1 fs1 fc1 sc0 ls18 ws2">3<span class="ff2 ls5d">x</span><span class="fs2 ls5e v1">1</span><span class="ff3 fs4 ls4 v0">+<span class="ff2 fs1 ls2">x</span></span><span class="fs2 ls2c v1">2</span><span class="ff3 fs4 lsb v0">=</span><span class="ls5f v0">3<span class="ff6 fs4 ls60 v8">)</span><span class="ff7 ls61 v14">\ue01a</span><span class="ff2 ls2 v15">x</span><span class="fs2 ls62 v16">1</span><span class="ff3 fs4 lsb v15">=</span></span><span class="v15">2</span></div><div class="t m0 x2f h1d y83 ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls2c v1">2</span><span class="ff3 fs4 ls63 v0">=<span class="ff6 ls64">\ue000<span class="ff1 fs1 ls18">3</span></span></span></div><div class="t m0 x8 h1e y84 ff1 fs1 fc1 sc0 ls18 ws1e">Geometricamente, as <span class="fc2 ws3a v0">retas <span class="fc1 ws13">rep<span class="_0 blank"></span>resentadas p<span class="_6 blank"> </span>elas duas equações acima</span></span></div><div class="t m0 x8 h4 y85 ff1 fs1 fc1 sc0 ls18 ws1f">se cruzam no p<span class="_6 blank"> </span>on<span class="_0 blank"></span>to<span class="_4 blank"> </span><span class="ff3 fs4 ls35 v0">(</span><span class="ws2 v0">2<span class="ff4 ls65">,<span class="ff6 fs4 ls66">\ue000</span></span><span class="ls67">3<span class="ff3 fs4 ls68 v0">)</span></span><span class="ff4">.</span></span></div><div class="t m0 x8 h4 y86 ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 2<span class="fc1">:</span></div><div class="t m0 x30 he y87 ff7 fs1 fc1 sc0 ls18">8</div><div class="t m0 x30 he y88 ff7 fs1 fc1 sc0 ls18"><</div><div class="t m0 x30 he y89 ff7 fs1 fc1 sc0 ls18">:</div><div class="t m0 x31 h1f y8a ff2 fs1 fc1 sc0 ls18 ws2">x<span class="ff1 fs2 ls69 v1">1</span><span class="ff6 fs4 ls4 v0">\ue000</span><span class="ff1 v0">0<span class="ff4 ls6a">,</span>4<span class="ff2">x</span><span class="fs2 ls6b v1">2</span><span class="ff6 fs4 ls6">\ue000</span>0<span class="ff4 ls6c">,</span>3<span class="ff2 ls2">x</span><span class="fs2 ls6d v1">3</span><span class="ff3 fs4 lsb">=</span>130</span></div><div class="t m0 x31 h4 y8b ff6 fs4 fc1 sc0 ls64">\ue000<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls6c">,</span>2<span class="ff2 ls2">x</span><span class="fs2 ls6e v1">1</span></span><span class="ff3 ls7">+<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls6a">,</span>88<span class="ff2 ls2">x</span><span class="fs2 ls6f v1">2</span></span></span><span class="ls4">\ue000<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls70">,</span>14<span class="ff2">x</span><span class="fs2 ls71 v1">3</span></span><span class="ff3 lsb">=<span class="ff1 fs1 ls18">74</span></span></span></div><div class="t m0 x31 h4 y8c ff6 fs4 fc1 sc0 ls64">\ue000<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls6c">,</span>5<span class="ff2 ls2">x</span><span class="fs2 ls6e v1">1</span></span><span class="ls7">\ue000<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls6a">,</span>2<span class="ff2 ls2">x</span><span class="fs2 ls72 v1">2</span></span><span class="ff3 ls4">+<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls73">,</span>95<span class="ff2">x</span><span class="fs2 ls71 v1">3</span></span><span class="lsb">=<span class="ff1 fs1 ls18">95</span></span></span></span></div><div class="t m0 x28 h20 y8d ff6 fs4 fc1 sc0 ls60">)<span class="ff7 fs1 ls18 v14">8</span></div><div class="t m0 x32 he y8e ff7 fs1 fc1 sc0 ls18"><</div><div class="t m0 x32 he y8f ff7 fs1 fc1 sc0 ls18">:</div><div class="t m0 x33 h21 y90 ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls2f v1">1</span><span class="ff3 fs4 lsb v0">=<span class="ff1 fs1 ls18">300</span></span></div><div class="t m0 x33 h4 y8b ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls74 v1">2</span><span class="ff3 fs4 lsb">=</span><span class="ff1 ls18">200</span></div><div class="t m0 x33 h22 y91 ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls74 v1">3</span><span class="ff3 fs4 lsb v0">=<span class="ff1 fs1 ls18">300</span></span></div><div class="t m0 x8 h23 y92 ff1 fs1 fc1 sc0 ls18 ws1e">Geometricamente, os <span class="fc2 ws3b v0">planos <span class="fc1 ws31">rep<span class="_0 blank"></span>resentados p<span class="_6 blank"> </span>elas três equações acima</span></span></div><div class="t m0 x8 h4 y93 ff1 fs1 fc1 sc0 ls18 ws1f">se cruzam no p<span class="_6 blank"> </span>on<span class="_0 blank"></span>to<span class="_4 blank"> </span><span class="ff3 fs4 ls35 v0">(</span><span class="ws2 v0">300<span class="ff4 ls75">,</span>200<span class="ff4 ls6c">,</span><span class="ws3c">300 <span class="ff3 fs4 ls35 v0">)</span>.</span></span></div><div class="t m0 x34 h5 y94 ff1 fs2 fc1 sc0 ls18 ws2c">10</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y22 ff1 fs0 fc0 sc0 ls18 ws3d">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as com V<span class="_0 blank"></span>ár<span class="_0 blank"></span>ias Sol<span class="_0 blank"></span>uçõ<span class="_0 blank"></span>es</div><div class="t m0 x8 h4 y95 ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 3<span class="fc1 ls76">:<span class="ff7 ls61 v11">\ue01a</span><span class="ff2 ls2 v12">x</span><span class="fs2 ls77 v13">1</span><span class="ff3 fs4 ls4 v12">+</span><span class="ls18 ws2 v12">2<span class="ff2 ls2">x</span></span><span class="fs2 ls14 v13">2</span><span class="ff3 fs4 lsb v12">=</span><span class="ls18 v12">3</span></span></div><div class="t m0 x35 h24 y96 ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2 ls2">x</span><span class="fs2 ls78 v1">1</span><span class="ff3 fs4 ls4">+</span>4<span class="ff2 ls2">x</span><span class="fs2 ls79 v1">2</span><span class="ff3 fs4 lsb">=</span><span class="ls5f">6<span class="ff6 fs4 ls7a v8">)</span></span><span class="ff2 v8">x</span><span class="fs2 ls7b v9">1</span><span class="ff3 fs4 lsb v8">=</span><span class="ls7c v8">3<span class="ff6 fs4 ls6">\ue000</span></span><span class="v8">2<span class="ff2">x</span></span><span class="fs2 v9">2</span></div><div class="t m0 x8 h4 y97 ff1 fs1 fc1 sc0 ls18 ws3e">Geometricamente, as retas rep<span class="_0 blank"></span>resentadas p<span class="_6 blank"> </span>elas duas equações acima</div><div class="t m0 x8 h4 y98 ff1 fs1 fc1 sc0 ls18 ws3f">são <span class="fc2 ws13 v0">pa<span class="_0 blank"></span>ralelas coincidentes<span class="fc1 ws1">, de forma que qualquer ponto sobre a reta</span></span></div><div class="t m0 x8 h4 y99 ff1 fs1 fc1 sc0 ls18 ws1">é solução do sistema, i.e.<span class="_2 blank"> </span>existem in\u2026nitas soluções.</div><div class="t m0 x8 h4 y9a ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 4<span class="fc1">:</span></div><div class="t m0 x36 he y9b ff7 fs1 fc1 sc0 ls5b">\ue01a<span class="ff2 ls2 v6">x</span><span class="ff1 fs2 ls7d v7">1</span><span class="ff3 fs4 ls4 v6">+</span><span class="ff1 ls18 ws2 v6">2<span class="ff2 ls2">x<span class="ff1 fs2 ls7e v1">2</span><span class="ff3 fs4 ls4">+</span></span>3<span class="ff2 ls2">x</span><span class="fs2 ls79 v1">3</span><span class="ff3 fs4 lsb">=</span>1</span></div><div class="t m0 x2 h25 y9c ff1 fs1 fc1 sc0 ls18 ws2">3<span class="ff2 ls5d">x</span><span class="fs2 ls7f v1">1</span><span class="ff3 fs4 ls4">+</span>2<span class="ff2 ls2">x</span><span class="fs2 ls15 v1">2</span><span class="ff3 fs4 ls4">+</span><span class="ff2 ls2">x</span><span class="fs2 ls2c v1">3</span><span class="ff3 fs4 lsb">=</span><span class="ls5f">1<span class="ff6 fs4 ls60 v8">)</span><span class="ff7 ls5b v14">\ue01a</span><span class="ff2 ls2 v15">x</span><span class="fs2 ls80 v16">1</span><span class="ff3 fs4 lsb v15">=</span><span class="ff2 ls2 v15">x</span></span><span class="fs2 v16">3</span></div><div class="t m0 x2f h26 y9d ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls81 v1">2</span><span class="ff3 fs4 lsb v0">=<span class="ff1 fs1 ls18 ws2">0<span class="ff4 ls75">,</span><span class="ls82">5</span></span><span class="ff6 ls4">\ue000<span class="ff1 fs1 ls18 ws2">2</span></span></span><span class="v0">x<span class="ff1 fs2 ls18 v1">3</span></span></div><div class="t m0 x8 h4 y9e ff1 fs1 fc1 sc0 ls18 ws40">Geometricamente, a solução do sistema acima é dada p<span class="_6 blank"> </span>ela intersecção</div><div class="t m0 x8 h27 y9f ff1 fs1 fc1 sc0 ls18 ws41">de dois planos e consiste em uma <span class="fc2 ws42 v0">reta <span class="fc1 ws43">no espaço <span class="ffa ls83">R</span><span class="fs2 ls84 ve">3</span><span class="ff4">.</span></span></span></div><div class="t m0 x34 h5 y30 ff1 fs2 fc1 sc0 ls18 ws2c">11</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls18 ws3d">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as sem S<span class="_0 blank"></span>olu<span class="_0 blank"></span>ção</div><div class="t m0 x8 h4 ya0 ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 5<span class="fc1 ls85">:<span class="ff7 ls61 v11">\ue01a</span><span class="ff2 ls18 ws2 v12">x</span><span class="fs2 ls86 v13">1</span><span class="ff3 fs4 ls4 v12">+</span><span class="ls18 ws2 v12">2<span class="ff2">x</span></span><span class="fs2 ls17 v13">2</span><span class="ff3 fs4 lsb v12">=</span><span class="ls18 v12">3</span></span></div><div class="t m0 x15 h28 ya1 ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2">x</span><span class="fs2 ls87 v1">1</span><span class="ff3 fs4 ls4 v0">+</span><span class="v0">4<span class="ff2">x</span><span class="fs2 ls88 v1">2</span><span class="ff3 fs4 lsb">=</span><span class="ls89">4<span class="ff6 fs4 ls8a v8">)</span></span><span class="ff8 ws44 v8">? ? ?</span></span></div><div class="t m0 x8 h4 ya2 ff1 fs1 fc1 sc0 ls18 ws3e">Geometricamente, as retas rep<span class="_0 blank"></span>resentadas p<span class="_6 blank"> </span>elas duas equações acima</div><div class="t m0 x8 h4 ya3 ff1 fs1 fc1 sc0 ls18 ws45">são <span class="fc2 ws46 v0">pa<span class="_0 blank"></span>ralelas não-coincidentes<span class="fc1 ws47">, de f<span class="_c blank"> </span>o<span class="_0 blank"></span>rma que não existe solução para o</span></span></div><div class="t m0 x8 h4 ya4 ff1 fs1 fc1 sc0 ls18 ws2">sistema.</div><div class="t m0 x8 h4 ya5 ff1 fs1 fc2 sc0 ls18 ws1e">Exemplo 6<span class="fc1 ls8b">:<span class="ff7 ls18 v11">8</span></span></div><div class="t m0 x19 he ya6 ff7 fs1 fc1 sc0 ls18"><</div><div class="t m0 x19 he ya7 ff7 fs1 fc1 sc0 ls18">:</div><div class="t m0 x15 h29 ya8 ff2 fs1 fc1 sc0 ls2">x<span class="ff1 fs2 ls77 v1">1</span><span class="ff3 fs4 ls4 v0">+<span class="ff1 fs1 ls18 ws2">3</span></span><span class="v0">x<span class="ff1 fs2 ls8c v1">2</span><span class="ff3 fs4 lsb">=</span><span class="ff1 ls18">1</span></span></div><div class="t m0 x15 h4 ya9 ff1 fs1 fc1 sc0 ls18 ws2">3<span class="ff2 ls2">x</span><span class="fs2 ls7e v1">1</span><span class="ff3 fs4 ls4 v0">+<span class="ff2 fs1 ls2">x</span></span><span class="fs2 ls74 v1">2</span><span class="ff3 fs4 lsb v0">=</span><span class="v0">1</span></div><div class="t m0 x15 h4 yaa ff1 fs1 fc1 sc0 ls18 ws2">2<span class="ff2 ls2">x</span><span class="fs2 ls7e v1">1</span><span class="ff3 fs4 ls4">+</span>3<span class="ff2 ls36">x</span><span class="fs2 ls8d v1">2</span><span class="ff3 fs4 lsb">=</span>1</div><div class="t m0 x37 h11 yab ff6 fs4 fc1 sc0 ls3b">)<span class="ff8 fs1 ls18 ws44">? ? ?</span></div><div class="t m0 x8 h4 yac ff1 fs1 fc1 sc0 ls18 ws13">Geometricamente, as três retas acima se interceptam em três p<span class="_c blank"> </span>ontos</div><div class="t m0 x8 h4 yad ff1 fs1 fc1 sc0 ls18 ws2">diferentes.</div><div class="t m0 x34 h5 y16 ff1 fs2 fc1 sc0 ls18 ws2c">12</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg5.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws48">So<span class="_0 blank"></span>b<span class="_0 blank"></span>re a G<span class="_0 blank"></span>eom<span class="_0 blank"></span>etr<span class="_0 blank"></span>ia da<span class="_0 blank"></span>s Eq<span class="_0 blank"></span>uaç<span class="_0 blank"></span>ões L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x8 h4 yae ff1 fs1 fc1 sc0 ls18 ws1">Observe que dado um sistema de equações linea<span class="_0 blank"></span>res, p<span class="_c blank"> </span>o<span class="_6 blank"> </span>demos</div><div class="t m0 x8 h4 yaf ff1 fs1 fc1 sc0 ls18 ws13">analisá-lo em termos das suas linhas ou das suas colunas.</div><div class="t m0 x2b h1b yb0 ff2 fs6 fc0 sc0 ls55">i<span class="ff4 ls56">.<span class="ff1 fc1 ls18 ws35">Ab o<span class="_0 blank"></span>rdagem<span class="_5 blank"> </span>p elas<span class="_5 blank"> </span><span class="fc2 ws49">linhas </span><span class="ws4a">do sistema:</span></span></span></div><div class="t m0 x38 h2a yb1 ff7 fs6 fc1 sc0 ls8e">\ue01a<span class="ff1 ls18 wsd v17">2<span class="ff2 ls8f">x<span class="ff6 fs7 ls90">\ue000</span><span class="ls91">y<span class="ff3 fs7 ls92">=</span></span></span>1</span></div><div class="t m0 x39 h1b yb2 ff2 fs6 fc1 sc0 ls93">x<span class="ff3 fs7 ls94">+</span><span class="ls91">y<span class="ff3 fs7 ls92">=</span><span class="ff1 ls18">5</span></span></div><div class="t m0 x2c h1b yb3 ff1 fs6 fc1 sc0 ls18 ws4a">Assim, a solução p<span class="_c blank"> </span>o<span class="_c blank"> </span>de ser vista como a intersecção entre duas retas:</div><div class="t m0 x34 h5 y7a ff1 fs2 fc1 sc0 ls18 ws2c">13</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y22 ff1 fs0 fc0 sc0 ls18 ws0">So<span class="_0 blank"></span>b<span class="_0 blank"></span>re a Ge<span class="_0 blank"></span>om<span class="_0 blank"></span>etri<span class="_0 blank"></span>a das E<span class="_0 blank"></span>qu<span class="_0 blank"></span>açõ<span class="_0 blank"></span>es Lin<span class="_0 blank"></span>ea<span class="_0 blank"></span>re<span class="_0 blank"></span>s</div><div class="t m0 x8 h4 yb4 ff1 fs1 fc1 sc0 ls18 ws2">(Cont.)</div><div class="t m0 x2d h1b yb5 ff2 fs6 fc0 sc0 ls18 ws34">ii <span class="ff4 ls57">.</span><span class="ff1 fc1 ws35">Ab o<span class="_0 blank"></span>rdagem<span class="_5 blank"> </span>p elas<span class="_5 blank"> </span><span class="fc2 ws4b">colunas </span><span class="ws4a">do sistema:</span></span></div><div class="t m0 x3a h2b yb6 ff2 fs6 fc1 sc0 ls95">x<span class="ff7 ls96 va">\ue014</span><span class="ff1 ls18 vc">2</span></div><div class="t m0 x3b h2c yb7 ff1 fs6 fc1 sc0 ls97">1<span class="ff7 ls98 v18">\ue015</span><span class="ff3 fs7 ls99 vc">+</span><span class="ff2 ls9a vc">y</span><span class="ff7 ls96 v18">\ue014</span><span class="ls18 v16">-1</span></div><div class="t m0 x3c h2c yb8 ff1 fs6 fc1 sc0 ls9b">1<span class="ff7 ls9c v18">\ue015</span><span class="ff3 fs7 ls9d vc">=</span><span class="ff7 ls9e v18">\ue014</span><span class="ls18 v16">1</span></div><div class="t m0 x3d h2c yb9 ff1 fs6 fc1 sc0 ls9f">5<span class="ff7 ls18 v18">\ue015</span></div><div class="t m0 x2c h1b yba ff1 fs6 fc1 sc0 ls18 ws4c">Neste caso, o p<span class="_0 blank"></span>roblema é descobrir qual combinação dos veto<span class="_0 blank"></span>res-coluna</div><div class="t m0 x2c h1b ybb ff1 fs6 fc1 sc0 ls18 ws4a">do lado esquerdo geram o veto<span class="_0 blank"></span>r coluna do lado direito:</div><div class="t m0 x34 h5 y9 ff1 fs2 fc1 sc0 ls18 ws2c">14</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res:<span class="_d blank"> </span>Ca<span class="_0 blank"></span>so Ge<span class="_0 blank"></span>ral 2<span class="_0 blank"></span>x2</div><div class="t m0 x8 h4 ybc ff1 fs1 fc1 sc0 ls18 ws1e">Considere o seguinte sistema com duas equações e duas incógnitas:</div><div class="t m0 x15 he ybd ff7 fs1 fc1 sc0 ls5b">\ue01a<span class="ff2 ls49 v6">a</span><span class="ff1 fs2 ls18 ws28 v7">11<span class="_7 blank"> </span></span><span class="ff2 ls36 v6">x</span><span class="ff1 fs2 lsa0 v7">1</span><span class="ff3 fs4 ls4 v6">+</span><span class="ff2 ls49 v6">a</span><span class="ff1 fs2 ls18 ws28 v7">12<span class="_7 blank"> </span></span><span class="ff2 ls18 ws2 v6">x</span><span class="ff1 fs2 ls7b v7">2</span><span class="ff3 fs4 lsb v6">=</span><span class="ff2 ls18 ws2 v6">b<span class="ff1 fs2 v1">1</span></span></div><div class="t m0 x3e h2d ybe ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_7 blank"> </span></span><span class="ls36">x<span class="ff1 fs2 lsa1 v1">1</span><span class="ff3 fs4 ls4 v0">+</span></span><span class="v0">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_7 blank"> </span></span><span class="ls18 ws2">x<span class="ff1 fs2 ls7b v1">2</span><span class="ff3 fs4 lsb">=</span>b<span class="ff1 fs2 v1">2</span></span></span></div><div class="t m0 x8 h4 ybf ff1 fs1 fc1 sc0 ls18 ws13">onde assume-se os pa<span class="_0 blank"></span>râmetros <span class="ff2 ls8">a</span><span class="fs2 ws28 v1">1 1<span class="_7 blank"> </span></span><span class="ff4 lsa2">,<span class="ff2 ls0">a</span></span><span class="fs2 ws28 v1">12<span class="_7 blank"> </span></span><span class="lsa3">,<span class="ff2 ls40">a</span></span><span class="fs2 ws28 v1">21<span class="_2 blank"> </span></span><span class="lsa4">e<span class="ff2 ls0">a</span></span><span class="fs2 ws28 v1">22<span class="_e blank"> </span></span><span class="ws1e">são estritamente</span></div><div class="t m0 x8 h4 yc0 ff1 fs1 fc1 sc0 ls18 ws4d">p ositivos.</div><div class="t m0 x8 h4 yc1 ff1 fs1 fc1 sc0 ls18 ws13">P<span class="_0 blank"></span>o<span class="_c blank"> </span>demos resolver este sistema<span class="_4 blank"> </span>através do <span class="fc2 ws4e">méto do<span class="_4 blank"> </span>da<span class="_5 blank"> </span>substituição</span>.</div><div class="t m0 x34 h5 y16 ff1 fs2 fc1 sc0 ls18 ws2c">15</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></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 x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg6.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res:<span class="_d blank"> </span>Ca<span class="_0 blank"></span>so Ge<span class="_0 blank"></span>ral 2<span class="_0 blank"></span>x2</div><div class="t m0 x8 h4 yc2 ff1 fs1 fc2 sc0 ls18 ws1e">P<span class="_0 blank"></span>asso 1<span class="fc1 ws1">:<span class="_2 blank"> </span>Resolva a primeira equação pa<span class="_0 blank"></span>ra <span class="ff2 ls2">x</span><span class="fs2 lsf v1">1</span>:</span></div><div class="t m0 x3f h2e yc3 ff2 fs1 fc1 sc0 ls18 ws2">x<span class="ff1 fs2 lsa5 v1">1</span><span class="ff3 fs4 lsa6 v0">+</span><span class="ls40 v5">a</span><span class="ff1 fs2 ws2c vc">12</span></div><div class="t m0 x40 h2f yc4 ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws4f v1">11<span class="_f blank"> </span></span><span class="ls2 vb">x</span><span class="ff1 fs2 ls2c vc">2</span><span class="ff3 fs4 lsa7 vb">=</span><span class="ls18 ws2 vd">b<span class="ff1 fs2 v1">1</span></span></div><div class="t m0 xb h2f yc4 ff2 fs1 fc1 sc0 ls0">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_10 blank"> </span></span><span class="ff6 fs4 ls2b vb">!</span><span class="ls2 vb">x</span><span class="ff1 fs2 ls74 vc">1</span><span class="ff3 fs4 lsa7 vb">=</span><span class="ls18 ws2 vd">b<span class="ff1 fs2 v1">1</span></span></div><div class="t m0 x29 h2f yc4 ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_e blank"> </span></span><span class="ff6 fs4 lsa6 vb">\ue000</span><span class="ls0 vd">a</span><span class="ff1 fs2 ls18 ws28 v19">1 2</span></div><div class="t m0 x41 h30 yc4 ff2 fs1 fc1 sc0 ls0">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_f blank"> </span></span><span class="ls2 vb">x</span><span class="ff1 fs2 ls18 vc">2</span></div><div class="t m0 x8 h4 yc5 ff1 fs1 fc2 sc0 ls18 ws1e">P<span class="_0 blank"></span>asso 2<span class="fc1 ws13">:<span class="_2 blank"> </span>Substituir a expressão acima na segunda equação e resolver</span></div><div class="t m0 x8 h4 yc6 ff1 fs1 fc1 sc0 ls18 ws50">pa<span class="_0 blank"></span>ra <span class="ff2 ws2">x</span><span class="fs2 ls1 v1">2</span>:</div><div class="t m0 x42 h31 yc7 ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws2c v1">21<span class="_b blank"> </span></span><span class="ff7 lsa8 v1a">\ue012</span><span class="ls18 ws2 v5">b<span class="ff1 fs2 v1">1</span></span></div><div class="t m0 xd h32 yc8 ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_e blank"> </span></span><span class="ff6 fs4 lsa9 vb">\ue000</span><span class="vd">a</span><span class="ff1 fs2 ls18 ws28 v19">1 2</span></div><div class="t m0 x22 h33 yc8 ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_f blank"> </span></span><span class="ls18 ws2 vb">x</span><span class="ff1 fs2 lsaa vc">2</span><span class="ff7 lsab v1b">\ue013</span><span class="ff3 fs4 ls4 vb">+</span><span class="vb">a</span><span class="ff1 fs2 ls18 ws28 vc">22<span class="_7 blank"> </span></span><span class="ls18 ws2 vb">x</span><span class="ff1 fs2 lsac vc">2</span><span class="ff3 fs4 lsb vb">=</span><span class="ls18 ws2 vb">b<span class="ff1 fs2 v1">2</span></span></div><div class="t m0 x43 h34 yc9 ff6 fs4 fc1 sc0 ls60">!<span class="ff3 ls35 v0">(</span><span class="ff2 fs1 ls40">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="ls0">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_b blank"> </span></span></span></span><span class="ls4">\ue000<span class="ff2 fs1 ls40">a<span class="ff1 fs2 ls18 ws28 v1">1 2<span class="_7 blank"> </span></span><span class="ls0">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_7 blank"> </span></span></span></span><span class="ff3 lsad v0">)</span><span class="ff2 fs1 ls2">x<span class="ff1 fs2 ls74 v1">2</span></span><span class="ff3 lsae">=<span class="lsaf v0">(</span><span class="ff2 fs1 ls40">a<span class="ff1 fs2 ls18 ws28 v1">1 1<span class="_7 blank"> </span></span><span class="ls18 ws2">b<span class="ff1 fs2 lsa5 v1">2</span></span></span></span>\ue000<span class="ff2 fs1 ls8">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_7 blank"> </span></span><span class="ls18 ws2">b<span class="ff1 fs2 lsb0 v1">1</span><span class="ff3 fs4 v0">)</span></span></span></span></div><div class="t m0 x34 h5 y7a ff1 fs2 fc1 sc0 ls18 ws2c">16</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y22 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res:<span class="_d blank"> </span>Ca<span class="_0 blank"></span>so Ge<span class="_0 blank"></span>ral 2<span class="_0 blank"></span>x2</div><div class="t m0 x8 h4 yca ff1 fs1 fc1 sc0 ls18 ws4e">Note<span class="_5 blank"> </span>que<span class="_4 blank"> </span>existem<span class="_4 blank"> </span>três<span class="_5 blank"> </span>p ossibilidades:</div><div class="t m0 x2b h1b ycb ff2 fs6 fc0 sc0 ls55">i<span class="ff4 ls56">.<span class="ff1 fc1 ls18 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsb1">a</span><span class="fs2 ws4f v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsb3">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff6 fs7 lsb4">6<span class="ff3 lsb5">=</span></span><span class="ws32">0, então p<span class="_c blank"> </span>o<span class="_c blank"> </span>demos utiliza<span class="_0 blank"></span>r as expressões acima</span></span></span></div><div class="t m0 x2c h35 ycc ff1 fs6 fc1 sc0 ls18 ws52">pa<span class="_0 blank"></span>ra <span class="fc2 ws53 v0">resolver <span class="fc1 ws54">para <span class="ff2 wsd">x</span><span class="fs2 lsb6 v1">1</span><span class="lsb7">e</span><span class="ff2 wsd">x</span><span class="fs2 ls19 v1">2</span><span class="ws4a">.<span class="_e blank"> </span>Neste caso, o sistema p<span class="_c blank"> </span>ossui uma única</span></span></span></div><div class="t m0 x2c h1b ycd ff1 fs6 fc1 sc0 ls18 wsd">solução.</div><div class="t m0 x2d h1b yce ff2 fs6 fc0 sc0 ls18 ws34">ii <span class="ff4 ls57">.</span><span class="ff1 fc1 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsb1">a</span><span class="fs2 ws4f v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls99">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsb3">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs7 lsb8">=</span><span class="ws55">0 e <span class="ff2 ls10">a</span><span class="fs2 ws28 v1">11<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsb9 v1">2</span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws28 v1">21<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsba v1">1</span><span class="ff3 fs7 ls92">=</span><span class="ws4a">0, então <span class="fc2 ws56">as duas equações</span></span></span></span></div><div class="t m0 x2c h35 ycf ff1 fs6 fc2 sc0 ls18 ws4a">são idênticas<span class="fc1 ws57 v0">.<span class="_e blank"> </span>(Prove!)<span class="_e blank"> </span>Neste caso, o sistema p<span class="_c blank"> </span>ossui in\u2026nitas soluções.</span></div><div class="t m0 x2e h1b yd0 ff2 fs6 fc0 sc0 ls18 ws36">iii <span class="ff4 ls58">.</span><span class="ff1 fc1 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsb1">a</span><span class="fs2 ws4f v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls99">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsb3">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs7 lsb8">=</span><span class="ws55">0 e <span class="ff2 ls10">a</span><span class="fs2 ws28 v1">11<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsb9 v1">2</span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws28 v1">21<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsba v1">1</span><span class="ff6 fs7 lsb4">6<span class="ff3 ls92">=</span></span><span class="ws4a">0, então <span class="fc2 ws56">as duas equações</span></span></span></span></div><div class="t m0 x2c h36 yd1 ff1 fs6 fc2 sc0 ls18 ws4a">são incompatíveis<span class="fc1 ws55 v0">.<span class="_e blank"> </span>Neste caso, o sistema não p<span class="_c blank"> </span>ossui solução.</span></div><div class="t m0 x34 h5 yd2 ff1 fs2 fc1 sc0 ls18 ws2c">17</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls18 ws58">Pr<span class="_0 blank"></span>ova</div><div class="t m0 x8 h4 yd3 ff1 fs1 fc2 sc0 ls18 ws18">Prop osição:<span class="_2 blank"> </span><span class="fc1 ws4d">Sup onha<span class="_5 blank"> </span>que<span class="_4 blank"> </span><span class="ff2 ls40">a</span><span class="fs2 ws28 v1">1 1<span class="_6 blank"> </span></span><span class="ff2 ls40">a</span><span class="fs2 ws28 v1">2 2<span class="_b blank"> </span></span><span class="ff6 fs4 ls4">\ue000</span><span class="ff2 ls40">a</span><span class="fs2 ws28 v1">1 2<span class="_7 blank"> </span></span><span class="ff2 ls0">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs4 ls27">=</span><span class="ws1e">0 e <span class="ff2 ls49">a</span><span class="fs2 ws28 v1">1 1<span class="_7 blank"> </span></span><span class="ff2 ws2">b</span><span class="fs2 lsbb v1">2</span><span class="ff6 fs4 ls4">\ue000</span><span class="ff2 ls40">a</span><span class="fs2 ws28 v1">21<span class="_7 blank"> </span></span><span class="ff2 ws2">b</span><span class="fs2 lsbc v1">1</span><span class="ff3 fs4 lsb">=</span>0.</span></span></div><div class="t m0 x8 h4 yd4 ff1 fs1 fc1 sc0 ls18 ws13">Prove que as duas equações do sistema são idênticas, ie.<span class="_2 blank"> </span>rep<span class="_0 blank"></span>resentam</div><div class="t m0 x8 h4 yd5 ff1 fs1 fc1 sc0 ls18 ws5">retas pa<span class="_0 blank"></span>ralelas coincidentes<span class="ff4">.</span></div><div class="t m0 x8 h4 yd6 ff1 fs1 fc2 sc0 ls18 ws59">Prova: <span class="fc1 ws13">Observe que as condições dadas fo<span class="_0 blank"></span>rmam o seguinte sistema:</span></div><div class="t m0 x44 h37 yd7 ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="ls40">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_b blank"> </span></span><span class="ff6 fs4 ls4 v0">\ue000</span></span><span class="v0">a<span class="ff1 fs2 ls18 ws28 v1">1 2<span class="_7 blank"> </span></span><span class="ls40">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs4 ls27">=</span><span class="ff1 ls18">0</span></span></span></div><div class="t m0 xf h4 yd8 ff2 fs1 fc1 sc0 ls0">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="ls18 ws2">b<span class="ff1 fs2 lsbd v1">2</span><span class="ff6 fs4 ls4">\ue000</span><span class="ls49">a</span><span class="ff1 fs2 ws2c v1">21<span class="_7 blank"> </span></span>b<span class="ff1 fs2 lsbe v1">1</span><span class="ff3 fs4 lsb">=</span><span class="ff1">0</span></span></div><div class="t m0 x8 h4 yd9 ff1 fs1 fc1 sc0 ls18 ws8">Resolvendo pa<span class="_0 blank"></span>ra <span class="ff2 ls40">a</span><span class="fs2 ws2c v1">2 1<span class="_e blank"> </span></span><span class="lsbf">e<span class="ff2 ls8">a</span></span><span class="fs2 ws28 v1">2 2<span class="_7 blank"> </span></span><span class="ws1">, obtemos:</span></div><div class="t m0 x20 h38 yda ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_e blank"> </span></span><span class="ff3 fs4 lsc0 v0">=</span><span class="ls40 v5">a</span><span class="ff1 fs2 ls18 ws28 vc">11<span class="_6 blank"> </span></span><span class="ls18 ws2 v5">b<span class="ff1 fs2 v1">2</span></span></div><div class="t m0 x45 h18 ydb ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x14 h16 ydc ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws2c v1">22<span class="_e blank"> </span><span class="ff3 fs4 v1c">=</span></span></div><div class="t m0 x46 h39 ydd ff2 fs1 fc1 sc0 ls40">a<span class="ff1 fs2 ls18 ws28 v1">12<span class="_b blank"> </span></span><span class="ff7 lsc1 v16">\ue010</span><span class="fs2 lsc2 v2">a</span><span class="ff1 fs8 ls18 ws5a v1d">11<span class="_6 blank"> </span></span><span class="fs2 lsc3 v2">b</span><span class="ff1 fs8 ls18 v1d">2</span></div><div class="t m0 x47 h3a yde ff2 fs2 fc1 sc0 lsc4">b<span class="ff1 fs8 lsc5 v1e">1</span><span class="ff7 fs1 ls18 v1f">\ue011</span></div><div class="t m0 x20 h3b ydf ff2 fs1 fc1 sc0 ls0">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_11 blank"> </span></span><span class="ff6 fs4 lsc6 vb">!</span><span class="ls49 vb">a</span><span class="ff1 fs2 ls18 ws28 vc">22<span class="_e blank"> </span></span><span class="ff3 fs4 lsc0 vb">=</span><span class="vd">a</span><span class="ff1 fs2 ls18 ws28 v19">12<span class="_6 blank"> </span></span><span class="ls18 ws2 vd">b<span class="ff1 fs2 v1">2</span></span></div><div class="t m0 x48 h18 ydf ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x8 h26 ye0 ff1 fs1 fc1 sc0 ls18 ws5b">Assim, substituíndo as exp<span class="_0 blank"></span>ressões ac<span class="_c blank"> </span>ima em <span class="ff2 ls40">a</span><span class="fs2 ws28 v1">2 1<span class="_6 blank"> </span></span><span class="ff2 ls5d">x</span><span class="fs2 lsc7 v1">1</span><span class="ff3 fs4 lsc8 v0">+<span class="ff2 fs1 ls40">a</span></span><span class="fs2 ws28 v1">2 2<span class="_7 blank"> </span></span><span class="ff2 ws2 v0">x</span><span class="fs2 ls7b v1">2</span><span class="ff3 fs4 lsb v0">=</span><span class="ff2 ws2 v0">b</span><span class="fs2 lsc9 v1">2</span><span class="ws2 v0">temos</span></div><div class="t m0 x8 h4 ye1 ff1 fs1 fc1 sc0 ls18 ws2">que:</div><div class="t m0 x31 h18 ye2 ff2 fs1 fc1 sc0 ls49">a<span class="ff1 fs2 ls18 ws2c v1">11<span class="_7 blank"> </span></span><span class="ls18 ws2">b<span class="ff1 fs2 v1">2</span></span></div><div class="t m0 x49 h3c ye3 ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 lsca v1">1</span><span class="ls2 vb">x</span><span class="ff1 fs2 lscb vc">1</span><span class="ff3 fs4 lsa9 vb">+</span><span class="ls0 vd">a</span><span class="ff1 fs2 ws28 v19">12<span class="_7 blank"> </span></span><span class="vd">b</span><span class="ff1 fs2 v19">2</span></div><div class="t m0 x4a h3d ye3 ff2 fs1 fc1 sc0 ls18 ws2">b<span class="ff1 fs2 lsca v1">1</span><span class="ls2 vb">x</span><span class="ff1 fs2 lscc vc">2</span><span class="ff3 fs4 lsb vb">=</span><span class="vb">b</span><span class="ff1 fs2 lscd vc">2</span><span class="ff6 fs4 ls7a vb">!</span><span class="ls0 vb">a</span><span class="ff1 fs2 ws28 vc">11<span class="_7 blank"> </span></span><span class="vb">x</span><span class="ff1 fs2 lsce vc">1</span><span class="ff3 fs4 ls4 vb">+</span><span class="ls8 vb">a</span><span class="ff1 fs2 ws28 vc">12<span class="_7 blank"> </span></span><span class="ls36 vb">x</span><span class="ff1 fs2 lscf vc">2</span><span class="ff3 fs4 lsb vb">=</span><span class="vb">b</span><span class="ff1 fs2 lsd0 vc">1</span><span class="ff5 fs5 v2">\ue004</span></div><div class="t m0 x34 h5 ye4 ff1 fs2 fc1 sc0 ls18 ws2c">18</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></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 x0 ye5 w1 h3e" alt="" src="https://files.passeidireto.com/5903d2bf-2f4e-4f07-8917-b39eb0a801a6/bg7.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y17 ff1 fs0 fc0 sc0 ls18 ws4">Si<span class="_0 blank"></span>stem<span class="_0 blank"></span>as d<span class="_0 blank"></span>e Equ<span class="_0 blank"></span>aç<span class="_0 blank"></span>ões Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>res:<span class="_d blank"> </span>Ca<span class="_0 blank"></span>so Ge<span class="_0 blank"></span>ral 2<span class="_0 blank"></span>x2</div><div class="t m0 x8 h4 ye6 ff1 fs1 fc2 sc0 ls18">Resumo:</div><div class="t m0 x2b h1b ye7 ff2 fs6 fc0 sc0 ls55">i<span class="ff4 ls56">.<span class="ff1 fc1 ls18 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsb1">a</span><span class="fs2 ws2c v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsb3">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff6 fs7 lsb4">6<span class="ff3 lsb5">=</span></span><span class="ws4a">0, então o sistema p<span class="_c blank"> </span>ossui uma <span class="fc2 ws5c">única solução</span></span></span></span></div><div class="t m0 x2c h1b ye8 ff1 fs6 fc1 sc0 ls18 ws35">dada<span class="_b blank"> </span>p or:</div><div class="t m0 x4b h3f ye9 ff2 fs6 fc1 sc0 ls18 wsd">x<span class="ff1 fs2 lsd1 v1">1</span><span class="ff3 fs7 lsd2">=</span><span class="ls10 v8">a</span><span class="ff1 fs2 ws28 v9">22<span class="_7 blank"> </span></span><span class="v8">b</span><span class="ff1 fs2 lsd3 v9">1</span><span class="ff6 fs7 ls94 v8">\ue000</span><span class="lsb2 v8">a</span><span class="ff1 fs2 ws28 v9">12<span class="_7 blank"> </span></span><span class="v8">b</span><span class="ff1 fs2 v9">2</span></div><div class="t m0 x9 h40 yea ff2 fs6 fc1 sc0 lse">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="lsb2">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls94">\ue000</span>a<span class="ff1 fs2 ls18 ws28 v1">12<span class="_7 blank"> </span></span><span class="ls10">a<span class="ff1 fs2 ls18 ws28 v1">21<span class="_12 blank"> </span></span><span class="ff1 lsd4 v8">e</span><span class="ls18 wsd v8">x</span><span class="ff1 fs2 lsd5 v9">2</span><span class="ff3 fs7 lsd6 v8">=</span><span class="lsd7 v15">a</span><span class="ff1 fs2 ls18 ws28 v16">11<span class="_7 blank"> </span></span><span class="ls18 wsd v15">b</span><span class="ff1 fs2 lsd3 v16">2</span><span class="ff6 fs7 ls99 v15">\ue000</span><span class="v15">a</span><span class="ff1 fs2 ls18 ws28 v16">21<span class="_7 blank"> </span></span><span class="ls18 wsd v15">b<span class="ff1 fs2 v1">1</span></span></span></span></div><div class="t m0 x27 h41 yea ff2 fs6 fc1 sc0 lsb3">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_7 blank"> </span></span><span class="lse">a<span class="ff1 fs2 ls18 ws28 v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls99">\ue000</span><span class="ls10">a<span class="ff1 fs2 ls18 ws28 v1">12<span class="_7 blank"> </span></span><span class="lsb2">a<span class="ff1 fs2 ls18 ws28 v1">21</span></span></span></span></div><div class="t m0 x2d h1b yeb ff2 fs6 fc0 sc0 ls18 ws34">ii <span class="ff4 ls57">.</span><span class="ff1 fc1 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsd7">a</span><span class="fs2 ws2c v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsd7">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs7 lsb8">=</span><span class="ws4a">0 e <span class="ff2 ls10">a</span><span class="fs2 ws28 v1">11<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsd3 v1">2</span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws28 v1">21<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsd8 v1">1</span><span class="ff6 fs7 lsb4">6<span class="ff3 ls92">=</span></span>0, então o sistema <span class="fc2 wsd">não</span></span></span></div><div class="t m0 x2c h1b yec ff1 fs6 fc2 sc0 ls18 ws35">p ossui<span class="_b blank"> </span>solução<span class="fc1">.</span></div><div class="t m0 x2e h1b yed ff2 fs6 fc0 sc0 ls18 ws36">iii <span class="ff4 ls58">.</span><span class="ff1 fc1 ws51">Se <span class="ff2 lse">a</span><span class="fs2 ws2c v1">11<span class="_7 blank"> </span></span><span class="ff2 lsd7">a</span><span class="fs2 ws2c v1">22<span class="_b blank"> </span></span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws2c v1">12<span class="_7 blank"> </span></span><span class="ff2 lsd7">a</span><span class="fs2 ws28 v1">21<span class="_4 blank"> </span></span><span class="ff3 fs7 lsb8">=</span><span class="ws4a">0 e <span class="ff2 ls10">a</span><span class="fs2 ws28 v1">11<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsd3 v1">2</span><span class="ff6 fs7 ls94">\ue000</span><span class="ff2 lsb2">a</span><span class="fs2 ws28 v1">21<span class="_7 blank"> </span></span><span class="ff2 wsd">b</span><span class="fs2 lsd8 v1">1</span><span class="ff3 fs7 ls92">=</span><span class="ws55">0, então o sistema p<span class="_c blank"> </span>ossui</span></span></span></div><div class="t m0 x2c h1b yee ff1 fs6 fc2 sc0 ls18 ws56">in\u2026nitas soluções <span class="fc1 ws38">dadas<span class="_b blank"> </span>p or:</span></div><div class="t m0 x4c h42 yef ff2 fs6 fc1 sc0 ls18 wsd">x<span class="ff1 fs2 lsd1 v1">1</span><span class="ff3 fs7 lsd9">=</span><span class="v8">b</span><span class="ff1 fs2 v9">1</span></div><div class="t m0 x24 h43 yf0 ff2 fs6 fc1 sc0 lsb1">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_e blank"> </span></span><span class="ff6 fs7 lsda v8">\ue000</span><span class="lsb2 v15">a</span><span class="ff1 fs2 ls18 ws28 v16">12</span></div><div class="t m0 x4d h44 yf0 ff2 fs6 fc1 sc0 lsb2">a<span class="ff1 fs2 ls18 ws28 v1">11<span class="_f blank"> </span></span><span class="lsdb v8">x</span><span class="ff1 fs2 ls18 v9">2</span></div><div class="t m0 x34 h5 yf1 ff1 fs2 fc1 sc0 ls18 ws2c">19</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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