The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 4.18: Graphing Systems of Linear Equations if

Lines Intersect, Are Parallel, or Coincide

The graphs of systems of linear equations may intersect, be parallel, or coincide. Each of these

variations has different meanings for the solution of the system, causing confusion for many

students.

1. Make sure that your students understand these terms: intersect, parallel, and coincide.
   Emphasize that the graphs of systems of linear equations may intersect, be parallel, or
   coincide.


2. Explain that when the graphs of the equations in a system intersect, the intersection is the
   solution to each equation in the system. See 4.17: ‘‘Graphing Systems of Linear Equations
   When Lines Intersect.’’


3. Explain that if the graphs of the equations do not intersect at one point, they may be parallel
   or they may coincide. If the lines are parallel, there is no solution. If the lines coincide, the
   graph is one line. In this case, every solution to the system of equations is on that line and
   the number of solutions is infinite.


4. Review the information on the worksheet with your students. Present the examples
   and graphs of these systems of equations:y= 2 x+1,y= 3 x, which has one solution;
   y= 3 x+4,y= 3 x−1, which has no solution because the lines are parallel; and
   y= 3 x+2, 2y= 6 x+4, which has an infinite number of solutions. Note the slope and
   y-intercept of each graph.

EXTRA HELP:

Be sure you have rewritten the equation correctly.

ANSWER KEY:

(1)No solution (2)One solution; (−3,−9) (3)Infinite number of solutions;

all are on the graph of

y= 2 x− 1

(4)One solution; (0, 0) (5)No solution (6)Infinite number of solutions;

all are on the graph of

y=−x+ 10

(7)One solution; (6, 2) (8)No solution

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(Challenge)Disagree. If two lines have the same slope and differenty-intercepts, there is no

solution. If two lines have the same slope and the samey-intercept, there is an infinite number

of solutions.

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172 THE ALGEBRA TEACHER’S GUIDE