##### Itemd

The system corresponding to this augmented matrix has three equations and five unknowns. Our first guess is there are infinitely many solutions. If we write out the equations corresponding to the augmented matrix, we find

\begin{equation*} \begin{alignedat}{5} x_1 \amp {}+{} \amp 2x_2 \amp \amp \amp \amp {}+{} \amp 3x_5 \amp {}={} \amp 2 \\ \amp \amp \amp x_3 \amp {}+{} \amp 2x_4 \amp {}-{} \amp x_5 \amp {}={} \amp -1 \\ \end{alignedat} \end{equation*}

since the third row of the augmented matrix does not restrict the solution space. From here, we see that there are infinitely many solutions: if we make any choice for the variables $$x_2\text{,}$$ $$x_4\text{,}$$ and $$x_5\text{,}$$ we can find values for $$x_1$$ and $$x_3$$ that give a solution.

in-context