The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & 2 & 3 & 1\\
2 & -1 & 1 & 2\\
3 & 1 & 1 & 4\\
0 & 5 & 2 & 1
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & 0 & 0 & 4/3\\
0 & \leading{1} & 0 & 1/3\\
0 & 0 & \leading{1} & -1/3\\
0 & 0 & 0 & 0
\end{bmatrix}.
\end{align*}
Since $r = n = 3$ and the system is consistent by Theorem RCLS, Theorem CSRN guarantees a unique solution, which is
\begin{align*}
x_1 &= 4/3\\
x_2 &= 1/3\\
x_3 &= -1/3.
\end{align*}