Definition:Simultaneous Equations/Solution

Definition

An ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called a solution to, or of, the system.

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 $\mathbb S$.

Also see

• Results about simultaneous equations can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): simultaneous equations
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): simultaneous equations
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): simultaneous equations