Definition:Simultaneous Equations

Definition

A system of simultaneous equations is a set of equations:

$\forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$

That is:

\(\ds \beta_1\)

\(=\)

\(\ds \map {f_1} {x_1, x_2, \ldots x_n}\)

\(\ds \beta_2\)

\(=\)

\(\ds \map {f_2} {x_1, x_2, \ldots x_n}\)

\(\ds \)

\(\cdots\)

\(\ds \)

\(\ds \beta_m\)

\(=\)

\(\ds \map {f_m} {x_1, x_2, \ldots x_n}\)

Linear Equations

A system of simultaneous linear equations is a set of equations:

$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$

That is:

\(\ds \beta_1\)

\(=\)

\(\ds \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n\)

\(\ds \beta_2\)

\(=\)

\(\ds \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n\)

\(\ds \)

\(\cdots\)

\(\ds \)

\(\ds \beta_m\)

\(=\)

\(\ds \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n\)

Solution

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 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 $\mathbb S$.

Consistency

A system of simultaneous equations:

$\forall i \in \set {1, 2, \ldots m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$

that has at least one solution is consistent.

If a system has no solutions, it is inconsistent.

Also see

• Definition:Gaussian Elimination

Sources

• 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Equations
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): simultaneous equations