Definition:Simultaneous Equations/Solution Set

Definition

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 $\mathbb X$

Sources

• 1987: A.G. Hamilton: A First Course in Linear Algebra ... (previous) ... (next): $1$: Gaussian Elimination