\begin{definition}[Definition:Simultaneous Equations/Linear Equations]
A '''system of simultaneous linear equations''' is a set of equations:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
That is:
{{begin-eqn}}
{{eqn | l = \beta_1
      | r = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n
}}
{{eqn | l = \beta_2
      | r = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n
}}
{{eqn | o = \cdots}}
{{eqn | l = \beta_m
      | r = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n
}}
{{end-eqn}}
\end{definition}