Special Cases of Systems of Equations

The following is an explanation of two special cases of systems of equations.  When solving by algebraic methods, either substitution or elimination, you find that you have “knocked out” both the x and the y variable and your resulting equation which contains only constant numbers is false like 18 = 7, then the answer to the system is “no solution” or the empty set.  The lines are parallel and therefore inconsistent.  On the other hand, if you are solving by algebraic methods and you find that you have “knocked out” both the x and the y variable and your resulting equation its true like 6 = 6 or 0 = 0, then the answer to the system is infinite solutions.  The lines are the same and therefore dependent.

Solve the system, 2x + 10y = 3 and x = 1 – 5y, by substitution.  Since the second equation is solved for x, I use that value and substitute it into the first equation:

      2(1 – 5y) + 10y = 3

      2 – 10y + 10y = 3

                          2 = 3 … which is false and therefore the system has no solution.

Solve the system, 6x + 9y = 6 and 2x + 3y = 2, by elimination.  I choose to multiply the second line by – 3 in order to eliminate the x variable.

      (– 3) 2x + 3y(– 3)  = (– 3) 2  … the second line now becomes – 6x – 9y =  – 6

      Now place the equations in column form add the two equations:

      6x + 9y = 6
   – 6x – 9y = –6
              0  =  0  … which is true and therefore the system has infinite solutions.

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 19-Sep-2007