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 & 1 & 2 & 6
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & 0 & 0 & 0\\
0 & \leading{1} & 0 & 0\\
0 & 0 & \leading{1} & 0\\
0 & 0 & 0 & \leading{1}
\end{bmatrix}.
\end{align*}
Since $n = 3$ and $r = 4 = n+1$, the last column is a pivot column and Theorem RCLS guarantees that the system is inconsistent. Thus, we see that the given system has no solution.