Definition:Simultaneous Equations/Linear Equations

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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.


Sources