Definition:Simultaneous Equations/Linear Equations
Definition
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
Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the linear equations in $\ds \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$.
Then $\tuple {x_1, x_2, \ldots, x_n}$ is referred to as a solution (to the system of simultaneous linear equations).
Matrix Representation
A system of simultaneous linear equations can be (and commonly is) expressed in its matrix representation:
- $\mathbf A \mathbf x = \mathbf b$
where:
$\quad \mathbf A = \begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {bmatrix}$, $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$, $\mathbf b = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$
are matrices.
Examples
Arbitrary System $1$
The system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x_1 - x_2 + x_3\) | \(=\) | \(\ds 2\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds 4 x_1 + x_2 - x_3\) | \(=\) | \(\ds 1\) |
has as its solution set:
\(\ds x_1\) | \(=\) | \(\ds -\dfrac 1 2\) | ||||||||||||
\(\ds x_2\) | \(=\) | \(\ds \dfrac 1 2\) | ||||||||||||
\(\ds x_3\) | \(=\) | \(\ds \dfrac 3 2\) |
Arbitrary System $2$
The system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x_1 + x_2\) | \(=\) | \(\ds 2\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x_1 + 2 x_2\) | \(=\) | \(\ds 3\) |
has no solutions.
Arbitrary System $3$
The system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x_1 - x_2 + x_3\) | \(=\) | \(\ds 2\) |
has as its solution set:
\(\ds x_1\) | \(=\) | \(\ds 1 - \dfrac t 3\) | ||||||||||||
\(\ds x_2\) | \(=\) | \(\ds \dfrac t 3\) | ||||||||||||
\(\ds x_3\) | \(=\) | \(\ds t\) |
where $t$ is any number.
Also see
- Results about simultaneous linear equations can be found here.
Sources
- 1954: A.C. Aitken: Determinants and Matrices (8th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $2$. Linear Equations and Transformations
- 1958: P.M. Cohn: Linear Equations ... (next): Introduction
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 30$. Linear Equations
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): simultaneous linear equations