# Definition:Simultaneous Equations/Linear Equations

## Definition

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:

 $\displaystyle \beta_1$ $=$ $\displaystyle \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n$ $\displaystyle \beta_2$ $=$ $\displaystyle \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n$ $\displaystyle$ $\cdots$ $\displaystyle$ $\displaystyle \beta_m$ $=$ $\displaystyle \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n$

### Solution

Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the equations in $\displaystyle \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$.

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 expressed as:

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

where:

$\mathbf A = \begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {bmatrix}$, $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf b = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$

are matrices.

## Examples

### Arbitrary System $1$

The system of simultaneous linear equations:

 $\text {(1)}: \quad$ $\displaystyle x_1 - 2 x_2 + x_3$ $=$ $\displaystyle 1$ $\text {(2)}: \quad$ $\displaystyle 2 x_1 - x_2 + x_3$ $=$ $\displaystyle 2$ $\text {(3)}: \quad$ $\displaystyle 4 x_1 + x_2 - x_3$ $=$ $\displaystyle 1$

has as its solution set:

 $\displaystyle x_1$ $=$ $\displaystyle -\dfrac 1 2$ $\displaystyle x_2$ $=$ $\displaystyle \dfrac 1 2$ $\displaystyle x_3$ $=$ $\displaystyle \dfrac 3 2$

### Arbitrary System $2$

The system of simultaneous linear equations:

 $\text {(1)}: \quad$ $\displaystyle x_1 + x_2$ $=$ $\displaystyle 2$ $\text {(2)}: \quad$ $\displaystyle 2 x_1 + 2 x_2$ $=$ $\displaystyle 3$

has no solutions.

### Arbitrary System $3$

The system of simultaneous linear equations:

 $\text {(1)}: \quad$ $\displaystyle x_1 - 2 x_2 + x_3$ $=$ $\displaystyle 1$ $\text {(2)}: \quad$ $\displaystyle 2 x_1 - x_2 + x_3$ $=$ $\displaystyle 2$

has as its solution set:

 $\displaystyle x_1$ $=$ $\displaystyle 1 - \dfrac t 3$ $\displaystyle x_2$ $=$ $\displaystyle \dfrac t 3$ $\displaystyle x_3$ $=$ $\displaystyle t$

where $t$ is any number.

## Also see

• Results about simultaneous linear equations can be found here.