Answer
Solution set: $\{(2,1,1)\}$
Work Step by Step
Begin with the system's augmented matrix.
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).
$\left[\begin{array}{lllll}
1 & 1 & 1 & | & 4\\
1 & -1 & -1 & | & 0\\
1 & -1 & 1 & | & 2
\end{array}\right]\left\{\begin{array}{l}
.\\
-R1.+R2\rightarrow R2\\
-R1+R3\rightarrow R3.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 1 & 1 & | & 4\\
0 & -2 & -2 & | & -4\\
0 & -2 & 0 & | & -2
\end{array}\right]\left\{\begin{array}{l}
.\\
\div(-2)\\
-R2+R3\rightarrow R3.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 1 & 1 & | & 4\\
0 & 1 & 1 & | & 2 \\
0 & 0 & 2 & | & 2
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\Rightarrow 2z=2.
\end{array}\right.$
$z=1$
Back substitute $z=2$ into row/equation 2,
$y+(1)=2$
$y=2-1=1$
Back substitute into row/equation 1:
$x+(1)+(1)=4$
$x=4-2=2$
Solution set: $\{(2,1,1)\}$
