Let $k$ be a field, let $m,n\in \mathbb{N}$, 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 of the linear system $Ax=b$. Then there is a finite sequence of elementary row operations transforming $[A\mid b]$ into row echelon form as an $(m\times(n+1))$ matrix, meaning that all nonzero rows lie above all zero rows, the leading nonzero entry of each nonzero row lies strictly to the right of the leading nonzero entry of the row above it, and every entry below a leading nonzero entry is zero. Moreover, after each elementary row operation in this sequence, the resulting augmented matrix defines a linear system with the same solution set in $k^n$ as the original system $Ax=b$.