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

       x1 + 2x2 - x3     = 4
      2x1 -  4x2           = 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 x1 and x2 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.