Definition:Simultaneous Equations

Definition

A system of simultaneous equations is a set of equations:

$\forall i \in \left[{1 \,.\,.\, m}\right] : f_i \left({x_1, x_2, \ldots x_n}\right) = \beta_i$

That is:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_1\)	\(=\)	\(\displaystyle f_1 \left({x_1, x_2, \ldots x_n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_2\)	\(=\)	\(\displaystyle f_2 \left({x_1, x_2, \ldots x_n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\cdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_m\)	\(=\)	\(\displaystyle f_m \left({x_1, x_2, \ldots x_n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Linear Equations

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

$\displaystyle \forall i \in \left[{1 \,.\,.\, m}\right] : \sum_{j=1}^n \alpha_{i j} x_j = \beta_i$

That is:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_1\)	\(=\)	\(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_2\)	\(=\)	\(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\cdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \beta_m\)	\(=\)	\(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Matrix Representation

A system of simultaneous equations can be expressed as:

$\mathbf A \mathbf x = \mathbf b$

where:

$ \mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf b = \begin{bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end{bmatrix}$

are matrices.

Solution

An $n$-tuple $\left({x_1, x_2, \ldots, x_n}\right)$ 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:

$\mathbb S := \forall i \in \left[{1 \,.\,.\, m}\right] : f_i \left({x_1, x_2, \ldots x_n}\right) = \beta_i$

Let $\mathbb X$ be the set of $n$-tuples:

$\left\{{\left\langle{x_j}\right\rangle_{j \in \left[{1 \,.\,.\, n}\right]}: \forall i \in \left[{1 \,.\,.\, m}\right]: f_i \left\langle{x_j}\right\rangle = \beta_i}\right\}$

which satisfies each of the equations in $\mathbb S$.

Then $\mathbb X$ is called the solution set of $\mathbb S$.

Consistency

A system that has at least one solution is said to be consistent.

If a system has no solutions, it is said to be inconsistent.

Also see

• Solution to Simultaneous Linear Equations

Sources

• Seth Warner: Modern Algebra (1965): $\S 30$