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.

in-context