Characterization of Parallel Lines

Examples, solutions, worksheets, and videos to help Grade 8 students learn that when a system of linear equations has no solution, i.e., no point of intersection of the lines, then the lines are parallel.

New York State Common Core Math Grade 8, Module 4, Lesson 26

Worksheets for Grade 8

Lesson 26 Student Outcomes

• Students know that when a system of linear equations has no solution, i.e., no point of intersection of the lines, then the lines are parallel.

Lesson 26 Student Summary

By definition, parallel lines do not intersect; therefore, a system of linear equations that graph as parallel lines will have no solution.
Parallel lines have the same slope, but no common point. Verify that lines are parallel by comparing their slopes and their y-intercepts.

Lesson 26 Opening Exercise

Exercises

1. Graph the system:
   y = 2/3 x + 4
   y = 4/6 x - 3
   a. Identify the slope of each equation. What do you notice?
   b. Identify the y-intercept of each equation. Are the y-intercepts the same or different?
2. Graph the system:
   y = -5/4 x + 7
   y = - 5/4 x + 2
   a. Identify the slope of each equation. What do you notice?
   b. Identify the y-intercept of each equation. Are the y-intercepts the same or different?
3. Graph the system:
   y = 2x - 5
   y = 2x - 1
   a. Identify the slope of each equation. What do you notice?
   b. Identify the y-intercept of each equation. Are the y-intercepts the same or different?

Theorem.
(1) Two distinct, non-vertical lines in the plane are parallel if they have the same slope.
(2) If two distinct, non-vertical lines have the same slope, then they are parallel.

4. Write a system of equations that has no solution
5. Write a system of equations that has (2, 1) as a solution.

6. How can you tell if a system of equations has a solution or not?
7. Does the system of linear equations shown below have a solution? Explain.
   6x - 2y = 5
   4x - 3y = 5
8. Does the system of linear equations shown below have a solution? Explain.
   -2x + 8y = 14
   x = 4y + 1
9. Does the system of linear equations shown below have a solution? Explain.
   12x + 3y = -2
   4x + y = 7
10. Genny babysits for two different families. One family pays her $6 each hour and a bonus of $20 at the end of the night. The other family pays her $3 every half hour and a bonus of $25 dollars at the end of the night. Write and solve the system of equations that represents this situation. At what number of hours do the two families pay the same for babysitting service from Genny?

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