Let $k$ be a field, let $A \in k^{m \times n}$, and let $b \in k^m$. Let $(A \mid b) \in k^{m \times (n+1)}$ denote the augmented matrix obtained by appending $b$ as the final column of $A$. Then the linear system $Ax=b$ has a solution $x \in k^n$ if and only if