Definition:Simultaneous Equations/Linear Equations/Solution

Definition
Consider the system of simultaneous linear equations:


 * $\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$

That is:

Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the equations in $\ds \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$.

Then $\tuple {x_1, x_2, \ldots, x_n}$ is referred to as a solution to the system of simultaneous linear equations'''