Definition:Simultaneous Equations

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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 linear 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 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 $\mathbb S$.


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.


Examples

Arbitrary Example

The simultaneous equations:

\(\text {(1)}: \quad\) \(\ds x + y\) \(=\) \(\ds 6\)
\(\text {(2)}: \quad\) \(\ds 2 x + y\) \(=\) \(\ds 4\)

has the solution:

\(\ds x\) \(=\) \(\ds -2\)
\(\ds y\) \(=\) \(\ds 8\)

This can be interpreted as that the point $\tuple {-2, 8}$ on the Cartesian plane is where the two straight lines $x + y = 6$ and $2 x + y = 4$ intersect.


Also see

  • Results about simultaneous equations can be found here.


Sources

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