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
x3 = 4
2x1 - 4x2 = 5
This system has the augmented matrix
which row reduces to
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 x1
are basic variables. The third column is not a
pivot column, so x3 is a free variable. Finally, the last column is not a pivot column, so the system is consistent.