#### Answer

The solution set is$\underline{\left( 4,0,-5 \right)}$.

#### Work Step by Step

The given equation set is\[\begin{align}
& 3x-2y+z=7 \\
& 2x+3y-z=13 \\
& x-y+2z=-6
\end{align}\].
Calculation:
The provided equations are,
\[3x-2y+z=7\] …(1)
\[2x+3y-z=13\]…(2)
\[x-y+2z=-6\]…(3)
Simplify as follows:
Step 1:
Let’s eliminate$z$and make two equations in two variables, use the addition method in equation (1) and (2) to eliminate the variable$z$,
\[3x-2y+z+\left( 2x+3y-z \right)=7+13\]
\[5x+y=20\] …(4)
And multiply by $2$ in equation (2), use the addition method in equation (2) and (3) to eliminate the variable$z$,
\[4x+6y-2z+\left( x-y+2z \right)=26+\left( -6 \right)\]
$5x+5y=20$ …(5)
Step 2:
Subtracting equation (5) from (4) to eliminate the variable$x$,
$\begin{align}
& 5x+y-\left( 5x+5y \right)=20-20 \\
& 5x+y-5x-5y=0 \\
& -4y=0 \\
& y=0
\end{align}$
Step 3:
Put the value of $y$ in equation (4) and simplify as follows,
$\begin{align}
& 5x+\left( 0 \right)=20 \\
& 5x=20 \\
& x=4
\end{align}$
Step 4:
Put the value of $x$ and $y$ in equation (1) and find out the value of$z$, simplify as follows:
\[\begin{align}
& 3x-2y+z=7 \\
& 3\left( 4 \right)-2\left( 0 \right)+z=7 \\
& 12+z=7 \\
& z=-5
\end{align}\]
Therefore, these three equations have a solution of$x=4,\,\,y=0\text{ and }z=-5$.