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The solution of a set of *m* linear equations in *n* unknowns can be investigated by the method of Gaussian elimination that transforms the augmented matrix to 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:(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.(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.(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 infinitely many solutions.When the set of equations is consistent, that is, in cases (ii) and (iii), the solution or solutions can be found either from the echelon form using back-substitution or by using Gauss–Jordan elimination to find the reduced echelon form. When there are infinitely many solutions, they can be expressed in terms of parameters that replace those unknowns free to take arbitrary values.

(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.

(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.

(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 infinitely many solutions.

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* The Concise Oxford Dictionary of Mathematics
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