Answer
$\{(1,3,-4)\}$.
Work Step by Step
The given system of equations is
$x+2y+3z=-5$
$2x+y+z=1$
$x+y-z=8$
The augmented matrix is
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 2 & 3 &-5\\
2 & 1 & 1 &1 \\
1 & 1 & -1 &8
\end{array}\right]$
Perform $R_2\rightarrow R_2-2 R_1$ and $R_3\rightarrow R_3- R_1$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 2 & 3 &-5\\
2-2(1) & 1-2(2) & 1-2(3) &1-2(-5) \\
1-1 & 1-2 & -1-3 &8-(-5)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 2 & 3 &-5\\
0 & -3 & -5 &11 \\
0 & -1 & -4 &13
\end{array}\right]$
Perform $R_2\rightarrow R_2/(-3)$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 2 & 3 &-5\\
0/(-3) & -3/(-3) & -5/(-3) &11/(-3) \\
0 & -1 & -4 &13
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 2 & 3 &-5\\
0 & 1 & 5/3 &-11/3 \\
0 & -1 & -4 &13
\end{array}\right]$
Perform $R_1\rightarrow R_1-2 R_2$ and $R_3\rightarrow R_3+ R_2$.
$\Rightarrow \left[\begin{array}{ccc|c}
1-2(0) & 2-2(1) & 3-2(5/3) &-5-2(-11/3)\\
0 & 1 & 5/3 &-11/3 \\
0+0 & -1+1 & -4+5/3 &13+(-11/3)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & -1/3 &7/3\\
0 & 1 & 5/3 &-11/3 \\
0 & 0 & -7/3 &28/3
\end{array}\right]$
Perform $R_3\rightarrow R_3(-3/7)$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & -1/3 &7/3\\
0 & 1 & 5/3 &-11/3 \\
0(-3/7) & 0(-3/7) & -7/3(-3/7) &28/3(-3/7)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & -1/3 &7/3\\
0 & 1 & 5/3 &-11/3 \\
0 & 0 & 1 &-4
\end{array}\right]$
Perform $R_1\rightarrow R_1+(1/3) R_2$ and $R_2\rightarrow R_2-(5/3) R_3$.
$\Rightarrow \left[\begin{array}{ccc|c}
1+(1/3)(0) & 0+(1/3)(0) & -1/3+(1/3)(1) &7/3+(1/3)(-4)\\
0-(5/3)(0) & 1-(5/3)(0) & 5/3-(5/3)(1) &-11/3-(5/3)(-4) \\
0 & 0 & 1 &-4
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & 0 & 0 &1\\
0 & 1 & 0 &3 \\
0 & 0 & 1 &-4
\end{array}\right]$
Use back substitution to solve the linear system.
$\Rightarrow x=1$
and
$\Rightarrow y=3$.
and
$\Rightarrow z=-4$.
The solution set is $\{(x,y,z)\}=\{(1,3,-4)\}$.
