The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & -2 & 1 & -1 & 3\\
2 & -4 & 1 & 1 & 2\\
1 & -2 & -2 & 3 & 1
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & -2 & 0 & 0 & 3\\
0 & 0 & \leading{1} & 0 & -2\\
0 & 0 & 0 & \leading{1} & -2
\end{bmatrix}.
\end{align*}
Thus, we see we have an equivalent system for any scalar $x_2$:
\begin{align*}
x_1 &= 3 + 2x_2\\
x_3 &= -2\\
x_4 &= -2.
\end{align*}
For this system, $n = 4$ and $r = 3$. Since it is a consistent system by Theorem RCLS, Theorem CSRN guarantees an infinite number of solutions.