Question

Consider the linear system:

x + 2y + 3z = a
x + 3y + 8z = b
x + 2y + 2z = c

where a, b and c are arbitrary constants. Find all solutions of this system.

(12)

Please assist.

Answer 1

From the first and the third equations it follows that $z=a-c$. Substituting to the second and the third equations we have

$x+3y=b-8a+8c$ and $x+2y=c-2a+2c=3c-2a$.

Hence $y=b-8a+8c-3c+2a=b-6a+5c$ and $x=3c-2a-2b+12a-10c=10a-2b-7c$

The answer is

$$x=10a-2b-7c$$ $$y=-6a+b+5c$$ $$z=a-c$$