Lesson 7-1

Graphing Systems of Equations

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Objectives

• Determine whether a system of linear equations has 0, 1, or infinitely many solutions

• Solve a system of equations by graphing

Vocabulary

• System of equations – two or more equations

• Consistent – a system of equations that has at least one ordered pair that satisfies both equations

• Inconsistent – a system of equations with no ordered pair that satisfies both equations

• Independent – a system of equations with exactly one solution

• Dependent – a system of equations that has an infinite number of solutions

System of Equalities

• Solutions of two linear equations result in:

y

x

y

x

y

x

No Solutions One Solution Infinite Solutions

Because (graphically): Lines are parallel Lines Intersect Same Line

Example 1a

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of andare parallel, there are no solutions.

Example 1bUse the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of andare intersecting lines, there is one solution.

Example 1cUse the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of andcoincide, there are infinitely many solutions.

Example 2a

The graphs of the equations coincide. There are infinitely many solutions of this system of equations.

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

Answer:

Example 2b

The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions of this system of equations.

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

Answer:

Example 3

Bicycling Tyler and Pearl went on a 20-kilometer bike ride that lasted 3 hours. Because there were many steep hills on the bike ride, they had to walk for most of the trip. Their walking speed was 4 kilometers per hour. Their riding speed was 12 kilometers per hour. How much time did they spend walking?

Words You have information about the amount of time spent riding and walking. You also know the rates and the total distance traveled.

Variables Let the number of hours they rode andthe number of hours they walked. Write a

system of equations to represent the situation.

Example 3 contEquations

The number ofhours riding plus

the number ofhours walking equals

the total number of hours of the trip.

The distancetraveled riding plus

the distancetraveled walking equals

the total distance of the trip.

r + w = 3

12r + 4w = 20

Example 3 contGraph the equations and .

The graphs appear to intersect at the point with the coordinates (1, 2). Check this estimate by replacing r with 1 and w with 2 in each equation.

Answer: Tyler and Pearl walked for 3 hours.

Summary & Homework

• Summary:

• Homework: – Pg 372 16-36 even

Graph Reveals

Intersecting Lines

Same Line

Parallel Lines

Solutions One Infinitely many none

Terminology Consistent and independent

Consistent and dependent

inconsistent