The solution of a set of m linear equations in n unknowns can be investigated by the method of Gaussian elimination (or Gauss-Jordan elimination) that transforms the augmented matrix to echelon form (or reduced echelon form). The number of non-zero rows in the echelon form cannot be greater than the number of unknowns, and three cases can be distinguished:

1. (i) If the echelon form has a row with all its entries zero except for a non-zero entry in the last place, then the set of equations is inconsistent.

2. (ii) If case (i) does not occur and, in the echelon form, the number of non-zero rows is equal to the number of unknowns, then the set of equations has a unique solution.

3. (iii) If case (i) does not occur and, in the echelon form, the number of non-zero rows is less than the number of unknowns, then the set of equations has more than one solution. In this case scalar parameters can be introduced for the variable of each column that does not contain a leading 1. The other variables can then be determined by backward substitution.