Recognizing Type of Solutions to Systems of Equations

Solving systems of equations using matrices is very helpful, as there are several tricks to be utilized to simplify the system quickly. The most common use of the Elementary Row Operations are to simplify equations into row echelon form.

Row Echelon Form Requirements:
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros, the first nonzero entry is a 1.
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

From row echelon form, we can find the type of solutions the system will have. In the following example, we can tell that the system will have infinitely many solutions from the row echelon form.

(b) 

                                        

                                  

 Since the bottom row would transfer into the equation 0 = 0, the other equation x + (5/2)y = -1/2 is the only relevant equation. Therefore, any number x will result in a number for y, so there are infinitely many solutions.

Here's another example:

            

                                                        

                                     

                              

If you switch the final two equations at the end of this example, it is in row echelon form. That final line would transfer into the equation 0+0+0=12. Obviously, this equation is incorrect, so the system as a whole is inconsistent and therefore has no solutions.

Sources:
http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx