The matrix equation

\begin{equation*} A \vec{x} = \vec{b} \end{equation*}

is equivalent to the linear system with the augmented matrix

\begin{equation*} \begin{pmatrix} \vphantom{\Big(} A & \vec{b} \, \end{pmatrix} = \begin{pmatrix} \vphantom{\Big(} \vec{a}_1 & \cdots & \vec{a}_n & \vec{b} \end{pmatrix}. \end{equation*}