A variable is a **basic variable** if it
corresponds to
a *pivot column*.
Otherwise, the variable is known as a **free**
**variable**.
In order to determine which variables are basic and which are free, it is
necessary to *row
reduce* the *augmented matrix*
to echelon
form.

For instance, consider the system of linear equations

x_{1} +
2x_{2} -
x_{3} = 4

2x_{1} -
4x_{2}
= 5

This system has the *augmented
matrix*

1 | 2 | -1 | 4 |

2 | -4 | 0 | 5 |

which row reduces to

1 | 2 | -1 | 4 |

0 | -8 | 2 | -3 |

This last matrix is in echelon
form, so we can identify the *pivot
positions* (the locations in red.) The first and second
columns are *pivot columns*, so variables x_{1}
and x_{2}
are basic variables. The third column is not a

*pivot
column*,
so x_{3} is a free variable. Finally, the last column is not
a *pivot
column*,
so the system is *consistent*.