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:

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:

Such a system is often 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.

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.