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Example: Non-homogeneous inconsistent system

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.