# 8.2: How Many Solutions?

Difficulty Level: At Grade Created by: CK-12

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.

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 y\begin{align*}y-\end{align*}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, {4x+2y=6y=2x+3\begin{align*}\begin{cases} 4x+2y = 6 \\ y = -2x+3 \\ \end{cases}\end{align*}.

8. A linear system has no solution if both equations have the same slope but different y\begin{align*}y-\end{align*}intercepts.

A linear system has infinitely many solutions if both equations have the same slope and same y\begin{align*}y-\end{align*}intercepts.

A linear system has one solution if the two equations have different slopes and different y\begin{align*}y-\end{align*}intercepts.

9. one solution

10. one solution

11. no solution

12. infinitely many solutions

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