Answer to Question #185568 in Algebra for AGIE

Question

The first of these equations plus the second equals the third: x+ y+ z=2 X + 2y+ Z = 3 2x + 3y + 2z = 5. The first two planes meet along a line. The third plane contains that line, because if x, y, z satisfy the first two equations then they also __ . The equations have infinitely many solutions (the whole line L). Find three solutions on L.  

Answer 1

We have given three equations,

$x+y+z =2 -----(1)$

$x+2y+z =3 ----(2)$

$2x+3y+2z = 5 -----(3)$

We can observe that,

$(1)+(2) = (3)$

If $x,y,z$ satisfy $(1),(2)$ they satisfy $(3)$ also.

So, (3) also contains the line of intersection of (1) and (2) planes.

$(2)- (1)$

This gives $y=1$ $\implies x+1+z = 2 \implies x+z = 1$

Let $x=0 \implies z=1$

Let $x=1 \implies z=0$

Let $x=-1 \implies z=2$

Solutions are $(0,1,1),(1,1,0){}and {}(-1,1,2)$