Escrever equações reduzidas na variável z da reta que passa por A(-1, 6, 3) e B(2,2,1).

A(-1,6,3) e B(2,2,1)

AB=B-A=(2,2,1)-(-1,6,3)

AB=(3,-4,-2)

Logo o ponto B(2,2,1)  resulta na equação:

   x=2+3t

r:y=2-4t 

z=1-2t

entao

t=(x-2)/3    t=(y-2)/-4  e t=(z-1)/-2  igualando valores de t temos:

( x - 2 )/3 = ( z - 1 )/- 2 ⇒ x = ( 7 - 3z )/2 
e 

( y - 2 )/- 4 = ( z - 1 )/- 2 ⇒ y - 2 = 2z - 2 ⇒ y = 2z 
Portanto; 

....{ x = ( 7 - 3z )/2 
r : {................. .............◄───R
....{ y = 2z 

\(\[\begin{align} & \text{A}\left( \text{ - 1 }\text{, 6 }\text{, 3 } \right)\text{ } \\ & \text{B}\left( \text{ 2 }\text{, 2 }\text{, 1 } \right) \\ & \text{AB = B - A }\left( \text{ 2 }\text{, 2 }\text{, 1 } \right)\text{ - }\left( \text{ - 1 }\text{, 6 }\text{, 3 } \right) \\ & \text{AB = }\left( \text{ 3 }\text{, - 4 }\text{, - 2 } \right) \\ & \text{ }\!\!~\!\!\text{ B}\left( \text{ 2 }\text{, 2 }\text{, 1 } \right)\text{:} \\ & \\ & \text{ x = 2 + 3}\text{.t} \\ & \text{y = 2 - 4}\text{.t} \\ & \text{ z = 1 - 2}\text{.t} \\ & \\ & \text{t = }\left( \text{ x - 2 } \right)\text{/3 }\text{, t = }\left( \text{ y - 2 } \right)\text{/- 4 } \\ & \text{t = }\left( \text{ z - 1 } \right)\text{/- 2 } \\ & \left( \text{ x - 2 } \right)\text{/3 = }\left( \text{ y - 2 } \right)\text{/- 4 } \\ & \left( \text{ z - 1 } \right)\text{/- 2 } \\ & \left( \text{ x - 2 } \right)\text{/3 = }\left( \text{ z - 1 } \right)\text{/- 2 x } \\ & \left( \text{ 7 - 3z } \right)\text{/2} \\ & \left( \text{ y - 2 } \right)\text{/- 4 = }\left( \text{ z - 1 } \right)\text{/- 2 y - 2 } \\ & \text{2z - 2 y = 2z} \\ & \\ & \text{x = }\left( \text{ 7 - 3z } \right)\text{/2} \\ & \text{y = 2z } \\ & \\ \end{align}\] \)