###### Item 1.2.5.1.c

The augmented matrix and its reduced row echelon form are

\begin{equation*}
\left[\begin{array}{rrrr|r}
1 \amp 2 \amp -5 \amp -1 \amp -3 \\
-2 \amp -2 \amp 6 \amp -2 \amp 4 \\
1 \amp 0 \amp -1 \amp 9 \amp -7 \\
0 \amp -1 \amp 2 \amp -1 \amp 4
\end{array}\right]
\sim
\left[\begin{array}{rrrr|r}
1 \amp 0 \amp -1 \amp 0 \amp 2 \\
0 \amp 1 \amp -2 \amp 0 \amp -3 \\
0 \amp 0 \amp 0 \amp 1 \amp -1 \\
0 \amp 0 \amp 0 \amp 0 \amp 0
\end{array}\right]\text{.}
\end{equation*}

This gives the equations

\begin{equation*}
\begin{aligned}
x_1 \amp = 2 + x_3 \\
x_2 \amp = -3 + 2x_3 \\
x_4 \amp = -1 - x_3 \\
\end{aligned}
\end{equation*}

showing that \(x_3\) is a free variable and \(x_1\text{,}\) \(x_2\text{,}\) and \(x_4\) are basic variables. There are therefore infinitely many solutions.