The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & -2 & 1 & -1 & 0\\
2 & -4 & 1 & 1 & 0\\
1 & -2 & -2 & 3 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
\leading{1} & -2 & 0 & 0 & 0\\
0 & 0 & \leading{1} & 0 & 0\\
0 & 0 & 0 & \leading{1} & 0
\end{bmatrix}.
\end{align*}
Thus, we see that the system is consistent (as predicted by Theorem HSC) and has an infinite number of solutions (as predicted by Theorem HMVEI). With a suitable choice of $x_2$, each solution can be written as
\begin{align*}
\colvector{2x_2\\ x_2 \\ 0 \\ 0}
\end{align*}