Answer
$\{(1,3,-2)\}$
Work Step by Step
We are given the system of equations:
$\begin{cases}
x-y+z=-4\\
2x-3y+4z=-15\\
5x+y-2z=12
\end{cases}$
Use the elimination method. Multiply the first equation by -4 and add it to the second equation; then multiply the first equation by 2 and add it to the third, to eliminate $z$:
$\begin{cases}
2x-3y+4z-4(x-y+z)=-15-4(-4)\\
5x+y-2z+2(x-y+z)=12+2(-4)
\end{cases}$
$\begin{cases}
2x-3y+4z-4x+4y-4z=-15+16\\
5x+y-2z+2x-2y+2z=12-8
\end{cases}$
$\begin{cases}
-2x+y=1\\
7x-y=4
\end{cases}$
Add the two equations to eliminate $y$ and determine $x$:
$-2x+y+7x-y=1+4$
$5x=5$
$x=1$
Determine $y$:
$7x-y=4$
$7(1)-y=4$
$y=3$
Determine $z$:
$x-y+z=-4$
$1-3+z=-4$
$-2+z=-4$
$z=-2$
The solution set of the system is:
$\{(1,3,-2)\}$
