Answer
$\{(1,3,-2)\}$
Work Step by Step
$\left\{\begin{array}{lllll}
x & -y & +z & =-4 & \\
2x & -3y & +4z & =-15 & /R_{2}=r_{2}-2r_{1}\\
5x & +y & -2z & =12 & /R_{3}=r_{3}-5r_{1}
\end{array}\right.$
... Eliminate $x$ in eq. 2 and 3
$\left\{\begin{array}{lllll}
x & -y & +z & =-4 & \\
& -y & +2z & =-7 & \\
& 6y & -7z & =32 & /R_{3}=r_{3}+6r_{2}
\end{array}\right.$
... Eliminate y in equation 3
$\left\{\begin{array}{lllll}
x & -y & +z & =-4 & (1)\\
& -y & +2z & =-7 & (2)\\
& & 5z & =-10 & (3)
\end{array}\right.$
$(3)\Rightarrow z=-2$
$(2) \Rightarrow y=2z+7=-4+7=3$
$(1)\Rightarrow x=y-z-4=3+2-4=1$
Solution set: $\{(1,3,-2)\}$
