\begin{definition}[Definition:Simultaneous Equations/Solution Set]
Consider the system of $m$ simultaneous equations in $n$ variables:
:$\mathbb S := \forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$
Let $\mathbb X$ be the set of ordered $n$-tuples:
:$\set {\sequence {x_j}_{j \mathop \in \set {1, 2, \ldots, n} }: \forall i \in \set {1, 2, \ldots, m}: \map {f_i} {\sequence {x_j} } = \beta_i}$
which satisfies each of the equations in $\mathbb S$.
Then $\mathbb X$ is called the '''solution set''' of $\mathbb S$.
Thus to '''solve''' a system of simultaneous equations is to find all the elements of $\mathbb X$
\end{definition}