We row-reduce the coefficient matrix of the system of equations,
\begin{align*}
\begin{bmatrix}
-1 & 5 & 0 & 0 \\
-2 & 5 & 5 & 2 \\
-3 & -1 & 3 & 1 \\
7 & 6 & 5 & 1
\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 the row-reduced version of the coefficient matrix is the $4\times 4$ identity matrix, $I_4$ (Definition IM byTheorem NMRRI, we know the coefficient matrix is nonsingular. According to Theorem NMUS we know that the system is guaranteed to have a unique solution, based only on the extra information that the coefficient matrix is nonsingular.