The augmented matrix for the given linear system and its row-reduced form are:
\begin{align*}
\begin{bmatrix}
1 & 4 & 3 & -1 & 5\\
1 & -1 & 1 & 2 & 6\\
4 & 1 & 6 & 5 & 9
\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 & \leading{1}
\end{bmatrix}.
\end{align*}
For this system, we have $n = 4$ and $r = 3$. However, with a pivot column in the last column we see that the original system has no solution by Theorem RCLS.