Teaching Notes 3.21: Solving Systems of Equations That Have
One Solution, No Solution, or an Infinite Number of Solutions
Solving systems of equations poses a special problem when students attempt to eliminate a
variable and find that both variables are eliminated. Accustomed to finding a specific value for a
variable, some students become confused as to what to do next.- Explain that many systems of equations have one solution, others have no solution, and still
others may have an infinite number of solutions. - Remind your students that systems of equations may be solved using the substitution
method, the addition-or-subtraction method, or themultiplication with addition-or-
subtraction method. Note that they should always try to choose the most efficient method
for solving a system of equations. Depending on the abilities of your students, you may wish
to review 3:20: ‘‘Solving Systems of Equations Using a Variety of Methods.’’ - Review the information and examples on the worksheet with your students. Emphasize that
if both variables are eliminated, the system of equations may have an infinite number of
solutions or it may have no solution. The answer depends on whether the equation is true
or false. A true statement indicates the system has an infinite number of solutions. A false
statement indicates the system has no solution.
EXTRA HELP:
Sometimes it is necessary to rewrite one or both equations to ‘‘line up the variables’’ when using
the addition-or-subtraction method or the multiplication with addition-or-subtraction method.ANSWER KEY:
(1)No solution (2)x=0;y= 4 (3)x=3;y=− 3 (4)Infinite number of solutions
(5)x=−2;y=− 3
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(Challenge)Jay is incorrect. If he solved the system of equations, he would find that both variables
would be eliminated, leaving a true statement: 0=0. This system has an infinite number of
solutions.x=1andy=2isoneofthem.
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