The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & 4 & 3 & -1 & 0\\
1 & -1 & 1 & 2 & 0\\
4 & 1 & 6 & 5 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
\leading{1} & 0 & 7/5 & 7/5 & 0\\
0 & \leading{1} & 2/5 & -3/5 & 0\\
0 & 0 & 0 & 0 & 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 suitable choices of $x_3$ and $x_4$, each solution can be written as
\begin{align*}
\colvector{-\frac{7}{5} x_3 - \frac{7}{5} x_4 \\ -\frac{2}{5} x_3 + \frac{3}{5} x_4 \\ x_3 \\ x_4}
\end{align*}