Systems of linear algebraic equations occur abundantly in most fields of science and engineering.

Examples are;

- structural analysis (civil engineering)
- heat conduction (mechanical and chemical engineering)
- analysis of power grids (electrical engineering)
- production planning (economics)
- regression analysis (statistics)

One of the most common sources of linear algebraic is from the discretisation of ordinary or partial differential equations or integral equations.

A linear equation is an equation that can be expressed as,

*a*_{0}x_{0} + a_{1}x_{1} + … = b
where

*a*_{0},

*a*_{1}, … and

*b* are constants.

A finite set of linear equations is called a

**system of linear equations** or a

**linear system**. A set of numbers

*s*_{0},

*s*_{1}, … is a solution to a linear system if and only if the substitutions

*a*_{0} =

*s*_{0},

*a*_{1} =

*s*_{1}, … satisfies every equation in the system.

A linear system of

*n* linear equations in

*n* variables