Answer
$(-1,2,-2)$
Work Step by Step
We are given the system:
$\begin{cases}
2x-y+2z=-8\\
x+2y-3z=9\\
3x-y-4z=3
\end{cases}$
We will use the addition method. Multiply Equation 1 by 2 and add it to Equation 2 to eliminate $y$. Also multiply Equation 3 by 2 and add it to Equation 2 to eliminate $y$:
$\begin{cases}
2(2x-y+2z)+x+2y-3z=2(-8)+9\\
2(3x-y-4z)+x+2y-3z=2(3)+9
\end{cases}$
$\begin{cases}
4x-2y+4z+x+2y-3z=-16+9\\
6x-2y-8z+x+2y-3z=6+9
\end{cases}$
$\begin{cases}
5x+z=-7\\
7x-11z=15
\end{cases}$
Multiply Equation 1 by 11 and add it to Equation 2 to eliminate $z$ and determine $x$:
$\begin{cases}
11(5x+z)=11(-7)\\
7x-11z=15
\end{cases}$
$\begin{cases}
55x+11z=-77\\
7x-11z=15
\end{cases}$
$55x+11z+7x-11z=-77+15$
$62x=-62$
$x=-1$
Substitute the value of $x$ in the Equation $5x+z=-7$ to determine $z$:
$5(-1)+z=-7$
$-5+z=-7$
$z=-7+5$
$z=-2$
Substitute the values of $x, z$ is Equation 1 of the given system to find $y$:
$2x-y+2z=-8$
$2(-1)-y+2(-2)=-8$
$-6-y=-8$
$y=-6+8$
$y=2$
The system's solution is:
$(-1,2,-2)$
