Answer
Solution set: $\{ ( 1$ , $2$ , $3$ , $-2 ) \}$
Work Step by Step
$\left[\begin{array}{llllll}
1 & 1 & 1 & 1 & | & 4\\
2 & 1 & -2 & -1 & | & 1\\
1 & -2 & -1 & -2 & | & -2\\
3 & 2 & 1 & 3 & | & -2
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftarrow-2R_{1}+R_{2}.\\
\leftarrow-R_{1}+R_{3}\\
\leftarrow-2R_{1}+R_{4}.
\end{array}\right.$
$\left[\begin{array}{llllll}
1 & 1 & 1 & 1 & | & 4\\
0 & -1 & -4 & -3 & | & -8\\
0 & -3 & -2 & -3 & | & -6\\
0 & -1 & -2 & 0 & | & -8
\end{array}\right]\left\{\begin{array}{l}
.\\
\times(-1).\\
\leftarrow 3R_{2}+R_{3}\\
\leftarrow R_{2}+R_{4}.
\end{array}\right.$
$\left[\begin{array}{llllll}
1 & 1 & 1 & 1 & | & 4\\
0 & 1 & 4 & 3 & | & 8\\
0 & 0 & 10 & 6 & | & 18\\
0 & 0 & 2 & 3 & | & 0
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
.\\
\leftarrow R_{3}-5R_{4}.
\end{array}\right.$
$\left[\begin{array}{llllll}
1 & 1 & 1 & 1 & | & 4\\
0 & 1 & 4 & 3 & | & 8\\
0 & 0 & 10 & 6 & | & 18\\
0 & 0 & 0 & -9 & | & 18
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
.\\
\Rightarrow z=-2.
\end{array}\right.$
Back substitute into row/equation 3,
$10y+6(-2)=18$
$10y=18+12=30$
$y=3$
Back substitute into row/equation 2:
$x+4(3)+3(-2)=8$
$x=8-6=2$
Back substitute into row/equation 1:
$w+2+3+(-2)=4$
$w=4-3=1$
Solution set: $\{ ( 1$ , $2$ , $3$ , $-2 ) \}$
