Answer
$(9, -4, -5)$ is a solution of the system
Work Step by Step
The second equation: $x+y+z=0$
Solve for x: $x=-y-z$
Substitute for x in the first equation:
$3x + 5y - z = 12 $
$3(-y-z) + 5y - z = 12 $
$-3y-3z+5y-z=12$
$2y-4z=12$ (*)
Substitute for x in the third equation:
$-x + 2y + 2z = -27$
$-(-y-z) + 2y + 2z = -27$
$y+z+2y+2z=-27$
$3y+3z=-27$
$\rightarrow 3y=-3z-27$
$\rightarrow y=-z-9$
Substitute for y:
$2y-4z=12$
$2(-z-9)-4z=12$
$-2z-18-4z=12$
$-6z=30$
$z=-5$
Solve for y: $y=-(-5)-9=-4$
Solve for x: $z=-(-4)-(-5)=9$
