We can reformulate the linear system as a vector equality with a matrix-vector product via Theorem SLEMM. The system is then represented by $A\vect{x}=\vect{b}$ where
\begin{align*}
A&=
\begin{bmatrix}
1 & -1 & 2 \\
1 & 0 & -2\\
2 & -1 & -1
\end{bmatrix}
&
\vect{b}
&=\colvector{5\\-8\\-6}
\end{align*}
According to Theorem SNCM, if $A$ is nonsingular then the (unique) solution will be given by $\inverse{A}\vect{b}$. We attempt the computation of $\inverse{A}$ through Theorem CINM, or with our favorite computational device and obtain,
\begin{align*}
\inverse{A}=
\begin{bmatrix}
2 & 3 & -2 \\
3 & 5 & -4 \\
1 & 1 & -1
\end{bmatrix}
\end{align*}
So by Theorem NI, we know $A$ is nonsingular, and so the unique solution is
\begin{align*}
\inverse{A}\vect{b}
=
\begin{bmatrix}
2 & 3 & -2 \\
3 & 5 & -4 \\
1 & 1 & -1
\end{bmatrix}
\colvector{5\\-8\\-6}
=
\colvector{-2\\-1\\3}
\end{align*}