The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & 2 & 3 & 0\\
2 & -1 & 1 & 0\\
1 & -8 & -7 & 0\\
0 & 1 & 1 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
\leading{1} & 0 & 1 & 0\\
0 & \leading{1} & 1 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}.
\end{align*}
An homogeneous system is always consistent (Theorem HSC) and with $n=3$, $r=2$ an application of Theorem FVCS yields one free variable. With a suitable choice of $x_3$ each solution can be written in the form
\begin{align*}
\colvector{-x_3 \\ -x_3 \\ x_3}
\end{align*}