\begin{definition}[Definition:Inconsistent Simultaneous Linear Equations]
A '''system of simultaneous linear equations''':
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
is referred to as '''inconsistent''' {{iff}} it has no solutions.
\end{definition}