Item1.2.5.1.a

The augmented matrix and its reduced row echelon form are

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

The linear system corresponding to the reduced row echelon form is

\begin{equation*} \begin{alignedat}{3} x \amp \amp \amp {}={} \amp -1 \\ \amp \amp y \amp {}={} \amp 2 \\ \amp \amp 0 \amp {}={} \amp 0\text{.} \\ \end{alignedat} \end{equation*}

This shows that there is a single solution \((-1,2)\) and that every variable is a basic variable.

in-context