# Definition:Simultaneous Equations

(Redirected from Definition:System of Equations)

Jump to navigation
Jump to search
## 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:

\(\displaystyle \beta_1\) | \(=\) | \(\displaystyle \map {f_1} {x_1, x_2, \ldots x_n}\) | |||||||||||

\(\displaystyle \beta_2\) | \(=\) | \(\displaystyle \map {f_2} {x_1, x_2, \ldots x_n}\) | |||||||||||

\(\displaystyle \) | \(\cdots\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \beta_m\) | \(=\) | \(\displaystyle \map {f_m} {x_1, x_2, \ldots x_n}\) |

### Linear Equations

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

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

That is:

\(\displaystyle \beta_1\) | \(=\) | \(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\) | |||||||||||

\(\displaystyle \beta_2\) | \(=\) | \(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\) | |||||||||||

\(\displaystyle \) | \(\cdots\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \beta_m\) | \(=\) | \(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\) |

## 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**.

## Sources

- 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Equations