Solve \(A \vec{x} = \vec{b}\), given

\begin{equation*} A = \begin{pmatrix} 1 & -2 & 3 \\ 2 & 1 & -1 \\ -3 & -4 & 5 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} 3 \\ 9 \\ -8 \end{pmatrix}. \end{equation*}

1. Write the augmented matrix and row-reduce it.

   \begin{align*} \begin{pmatrix} \vphantom{\Big(} A & \vec{b} \, \end{pmatrix} & = \begin{pmatrix} 1 & -2 & 3 & 3 \\ 2 & 1 & -1 & 9 \\ -3 & -4 & 5 & -8 \end{pmatrix} \\[2ex] \operatorname{rref} \begin{pmatrix} \vphantom{\Big(} A & \vec{b} \, \end{pmatrix} & = \begin{pmatrix} 1 & 0 & \frac{1}{5} & 0 \\ 0 & 1 & -\frac{7}{5} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{align*}

   The last column is the pivot column, so the system is inconsistent.