Answer
Solution set: $\{(1,-1,1)\}$
Work Step by Step
Begin with the system's augmented matrix.
$\left[\begin{array}{lllll}
1 & -2 & -1 & | & 2\\
2 & -1 & 1 & | & 4\\
-1 & 1 & -2 & | & -4
\end{array}\right]$
Use matrix row operations to get ls down the main diagonal from upper left to lower right, and Os below the $1\mathrm{s}$ (row-echelon form).
$\left[\begin{array}{lllll}
1 & -2 & -1 & | & 2\\
2 & -1 & 1 & | & 4\\
-1 & 1 & -2 & | & -4
\end{array}\right]\left\{\begin{array}{l}
.\\
-2R1+R2\rightarrow R2\\
R1+R3\rightarrow R3
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & -2 & -1 & | & 2\\
0 & 3 & 3 & | & 0\\
0 & -1 & 3 & | & -2
\end{array}\right]\left\{\begin{array}{l}
.\\
\times(-\frac{1}{3})\\
3R2+R3\rightarrow R3
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & -2 & -1 & | & 2\\
0 & 1 & 1 & | & 0\\
0 & 0 & -2 & | & -2
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\Rightarrow-2z=-2
\end{array}\right.$
$z=1$
Back substitute $z=1$ into row 2,
$y+z=0$
$y=-1$
Back substitute into equation 1:
$x-2(-1)-(1)=2$
$x=2-2+1=1$
Solution set: $\{(1,-1,1)\}$
