Answer
$\{(3,-1,-1)\}$.
Work Step by Step
The given system of equations is
$x+3y=0$
$x+y+z=1$
$3x-y-z=11$
The augmented matrix is
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 3 & 0& 0\\
1 &1 & 1& 1 \\
3&-1&-1&11
\end{array}\right]$
Perform $R_2\rightarrow R_2-R_1$ and $R_3\rightarrow R_3-3( R_1)$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 3 & 0& 0\\
1-1 &1 -3& 1-0& 1-0 \\
3-3(1)&-1-3(3)&-1-3(0)&11-3(0)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 3 & 0& 0\\
0 &-2& 1& 1 \\
0&-10&-1&11
\end{array}\right]$
Perform $R_2\Rightarrow R_2/(-2)$
Change the sign of the second row and swap the 3rd and 2nd row.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 3 & 0& 0\\
0/(-2) &-2/(-2)& 1/(-2)& 1/(-2) \\
0&-10&-1&11
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 3 & 0& 0\\
0 &1& -1/2& -1/2 \\
0&-10&-1&11
\end{array}\right]$
Perform $R_1\rightarrow R_1-3\times R_2$ and $R_3\rightarrow R_3+10 R_2$.
$\Rightarrow \left[\begin{array}{ccc|c}
1-3(0) & 3-3(1) & 0-3(-1/2)& 0-3(-1/2)\\
0 &1& -1/2& -1/2 \\
0+10(0)&-10+10(1)&-1+10(-1/2)&11+10(-1/2)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & 3/2& 3/2\\
0 &1& -1/2& -1/2 \\
0&0&-6&6
\end{array}\right]$
Perform $R_3\rightarrow \frac{R_3}{-6}$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & 3/2& 3/2\\
0 &1& -1/2& -1/2 \\
0/(-6) & 0 /(-6)& -6/(-6)& 6/(-6)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & 3/2& 3/2\\
0 &1& -1/2& -1/2 \\
0 & 0& 1& -1
\end{array}\right]$
Perform $R_1\rightarrow R_1-(3/2)R_3$ and $R_2\rightarrow R_2+(1/2) R_3$.
$\Rightarrow \left[\begin{array}{ccc|c}
1-(3/2)0 & 0-(3/2)0 & 3/2-(3/2)1& 3/2-(3/2)(-1)\\
0+(1/2)0 &1+(1/2)0& -1/2+(1/2)1& -1/2+(1/2)(-1) \\
0 & 0& 1& -1
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & 0& 3\\
0 &1& 0& -1 \\
0 & 0& 1& -1
\end{array}\right]$
Use back substitution to solve the linear system.
$\Rightarrow x=3$
and
$\Rightarrow y=-1$.
and
$\Rightarrow z=-1$.
The solution set is $\{(x,y,z)\}=\{(3,-1,-1)\}$.
