Definition:Simultaneous Equations

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:

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

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.