\begin{definition}[Definition:Equivalent Systems of Simultaneous Linear Equations]
Let $S_1$ and $S_2$ be two systems of simultaneous linear equations.
Then $S_1$ and $S_2$ are '''equivalent''' {{iff}}:
:every solution to $S_1$ is also a solution to $S_2$
and:
:every solution to $S_2$ is also a solution to $S_1$.
\end{definition}