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:

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.

Consistency
A system that has at least one solution is said to be consistent.

If a system has no solutions, it is said to be inconsistent.