Definition:Simultaneous Equations

Definition
A system of simultaneous equations is a set of equations:


 * $\forall i \in \left[{1 \, . \, . \, m}\right] : f_i \left({x_1, x_2, \ldots x_n}\right) = \beta_i$

That is:

Linear Equations
A system of simultaneous linear equations is a set of equations:


 * $\displaystyle \forall i \in \left[{1 \, . \, . \, m}\right] : \sum_{j=1}^n \alpha_{i j} x_j = \beta_i$

That is:

Solution
An $n$-tuple $\left({x_1, x_2, \ldots, x_n}\right)$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called a solution of the system.