The basic problem of linear algebra is to solve a system of linear equations. **A linear** equation in the *n* variables—or unknowns— *x* _{1}, *x* _{2}, …, and *x _{n} *is an equation of the form

where *b* and the coefficients *a _{i} *are constants. A finite collection of such linear equations is called a

**linear system**. To

**solve a system**means to find all values of the variables that satisfy all the equations in the system

*simultaneously*. For example, consider the following system, which consists of two linear equations in two unknowns:

Although there are infinitely many solutions to each equation separately, there is only one pair of numbers *x* _{1} and *x* _{2} which satisfies both equations at the same time. This ordered pair, (*x* _{1}, *x* _{2}) = (2, 1), is called the **solution** to the system.