8.2: How Many Solutions?
This activity is intended to supplement Algebra I, Chapter 7, Lesson 5.
ID: 9283
Time required: 60 minutes
Topic: Linear Systems
- Graph a system of linear equations to determine whether they have no solutions, one or infinitely many.
- Examine the coefficients of a pair of linear equations to determine how many solutions for the system.
Activity Overview
In this activity, students graph systems of linear functions to determine the number of solutions. Once acquainted with each of the three possibilities—one solution, zero solutions and infinitely many solutions—they use their experience with the graphs to investigate the relationship between the coefficients of a pair of linear equations and the number of solutions. In the investigation, students are given one line and challenged to draw a second line that creates a system with a particular number of solutions. By repeating this experiment and recording the equations of the line, students gather data that they use to write rules about the number of solutions of a linear system based on its coefficients.
Teacher Preparation
- This activity is appropriate for students in Algebra 1. It is assumed that students are familiar with linear functions, their graphs, and have solved linear systems algebraically.
- To download Cabri Jr. and its files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=9283, and select Cabri Jr., HOWMANY1, HOWMANY2, HOWMANY3, and HOWMANY4. If your calculator already has the Cabri Jr. application, you may still need to download the .8xv files.
Classroom Management
- This activity is designed to have students explore individually and in pairs. However, an alternate approach would be to use the activity in a whole-class format. By using the computer software and the questions found on the student worksheet, you can lead an interactive class discussion on the slope of perpendicular lines.
- The student worksheet guides students through the main ideas of the activity. You may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion.
Associated Materials
- Student Worksheet: How Many Solutions?, http://www.ck12.org/flexr/chapter/9617, scroll down to the second activity.
- Cabri Jr. Application
- HOWMANY1.8xv, HOWMANY2.8xv, HOWMANY3.8xv, HOWMANY4.8xv
Solutions
1.
one solution
2.
no solutions
3.
infinitely many solutions
4. The equations that form systems with no solutions or infinitely many solutions have the same slope.
5. The equations that form systems with infinitely many solutions have the same intercept.
6. The equations that form systems with infinitely many solutions are equivalent.
7. Sometimes equations are written differently although they equivalent. One equation may have been multiplied by a constant on both sides or terms moved around. For example, .
8. A linear system has no solution if both equations have the same slope but different intercepts.
A linear system has infinitely many solutions if both equations have the same slope and same intercepts.
A linear system has one solution if the two equations have different slopes and different intercepts.
9. one solution
10. one solution
11. no solution
12. infinitely many solutions