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\\
3 & 1 & 1 & 0\\
0 & 5 & 2 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}.
\end{align*}
An homogeneous system is always consistent (Theorem HSC) and with $n=r=3$ an application of Theorem FVCS yields zero free variables. Thus the only solution to the given system is the trivial solution, $\vect{x} = \zerovector$.