Graphing Systems of Equations - MR. JONES · 2020. 1. 7. · Learn Solving Systems of Equations by Graphing You can solve a system of equations by graphing each equation carefully

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Go Online You may want to complete the Concept Check to check your understanding.

Graphing Systems of Equations

Lesson 7-1

Go Online You can complete an Extra Example online.

Today’s Vocabulary system of equationsconsistentindependentdependentinconsistent

Explore Intersections of Graphs

Online Activity Use graphing technology to complete the Explore.

Learn Graphs of Systems of EquationsA set of two or more equations with the same variables is called a system of equations. An ordered pair that is a solution of both equations is a solution of the system. A system of two linear equations can have one solution, an infinite number of solutions, or no solution.

• A system of equations is consistent if it has at least one ordered pair that satisfies both equations.

• If a consistent system of equations has exactly one solution, it is said to be independent. The graphs intersect at one point.

• If a consistent system of equations has an infinite number of solutions, it is dependent. The graphs are the same line. This means that there are unlimited solutions that satisfy both equations.

• A system of equations is inconsistent if it has no ordered pair that satisfies both equations. The graphs are parallel.

Example 1 Consistent SystemsUse the graph to determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent.

y = −3x + 1y = x − 3

y

xO

y = -3x + 1y = x - 3

Since the graphs of these two lines intersect at one point, there is exactly one solution. Therefore, the system is .

INQUIRY How can you solve a linear equation by graphing?

Today’s Goals● Determine the number

of solutions of a system of linear equations.

● Solve systems of equations by graphing.

● Solve linear equations by graphing systems of equations.

● Use graphing calculators to solve systems of equations.

Lesson 7-1 • Graphing Systems of Equations 387

consistent and independent

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

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Study Tip

Parallel Lines Two lines that have the same slope are parallel and never intersect. A system of parallel lines has no solution.

Example 2 Inconsistent SystemsUse the graph to determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent.

y = 1 __ 2 x + 2

y = 1 __ 2 x − 1

y

xO

y = x + 212

y = x - 112

Since the graphs of these two lines are parallel, there is no solution of the system. Therefore, the system is .

CheckDetermine whether each graph shows a system that is consistent or inconsistent and if it is independent or dependent.

y

xO

y = - x - 212

y = - x + 312

y

xO

y = -x - 1

y = x - 4

Example 3 Number of Solutions, Equations in Slope-Intercept FormDetermine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent.

y = 6x + 10 Because the slopes are the and the y-intercepts are , the lines are

.

y = 6x + 4 The system has solution. Therefore, the system is .

CheckDetermine the number of solutions the system has.

y = 4 __ 5 x − 2

4y - 5x = 9


Think About It!

How could you change the equations so they form a consistent and dependent system?

Your Notes

388 Module 7 • Systems of Linear Equations and Inequalities

Sample answer: Change the value of b in the second equation to 2.

inconsistent consistent and independent

inconsistent

one solution

samedifferent

parallelno

inconsistent


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Think About It!

How many solutions will a system have if the slopes and y-intercepts of the lines are different?

Example 4 Number of Solutions, Equations in Standard FormDetermine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent.

4y − 6x = 169x − 6y = −24

Write both equations in slope-intercept form.

4y − 6x = 16

4y − 6x + 6x = + 16

4y =

4y

__ 4 = 6x __ 4 + 16 __ 4

y = 3 __ 2 x +

Original equation

Isolate the y-term

Simplify.

Divide by coefficient of y.

Simplify.

9x − 6y = −24

9x − 6y − 9x = −24

−6y =

−6y

___ −6 = −9x ___ −6 + −24 ____ −6

y = + 4

Because the slopes are and the y-intercepts are , this is line.

Since the graphs of these two lines are the same, there are infinitely many solutions. Therefore, the system is .

CheckDetermine the number of solutions each system has.

4x − 8y = 16

6x − 12y = 5

Learn Solving Systems of Equations by GraphingYou can solve a system of equations by graphing each equation carefully on the same coordinate plane. Every point that lies on the line of one equation represents a solution of that equation. Similarly, every point on the line of the second equation in a system represents a solution of that equation. Therefore, the solution of a system of equations is the point at which the graphs intersect.

For example, the solution of this system is (−1, 3). That is the point at which the graphs intersect. Since the point of intersection lies on both lines, the ordered pair satisfies each equation in the system.

y

xO

y = x + 4(-1, 3)

y = -2x + 1


exactly one

4

−9x−9x −24

+6x6x + 16

the same the samethe same

consistent and dependent

no solution

3 __ 2 x


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Think About It!

Why is it necessary to substitute the values of x and y into both equations to check your solution?

Go Online You can watch a video to see how to use a graphing calculator with this example.


Example 5 Solve a System by GraphingGraph the system and determine the number of solutions that it has. If it has one solution, determine its coordinates.

y = −2x + 14y = 3 __ 5 x + 1

y

xO−2−6−10

2

6

10

14

2 6 10

−6

The graphs of the lines appear to intersect at the point ( ). If you substitute 5 for x and 4 for y into the equations, both are true. Therefore, is the solution of the system.

CheckGraph the system of equations.

3x + 5y = 10x − 5y = −10

What is the solution of the system?

( )

y

xO 84−8 −4

−8

−4

4

8

Example 6 Graph and Solve a System of EquationsGraph the system and determine the number of solutions that it has. If it has one solution, determine its coordinates.

−3x + 2y = 126x − 4y = 8

y

xO

−10

−2−6−10

−6

2

6

10

2 6 10

The lines have the same slope but different y-intercepts, so the lines are parallel. Since they do not intersect, this system has .


Sample answer: The point must be a solution of both equations in order to be a solution of the system.

0, 2

5, 4

(5, 4)

no solution


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Apply Example 7 Write a System of EquationsPOPULATION China and India are the two most populous countries in the world. The populations of these countries have increased steadily in recent years. In 2010, China and India had populations of about 1.34 billion and 1.19 billion respectively. By 2016, the populations grew to about 1.38 billion in China and 1.29 billion in India. Predict the approximate year when the populations of the two countries will be the same.

1. What is the task?Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions?

2. How will you approach the task? What have you learned that you can use to help you complete the task?

3. What is your solution? Use your strategy to solve the problem.Find the average rate of change for the populations of China and India.

China:

India: Write a system of equations to represent the system.

Study Tip

Assumptions Populations do not typically increase at a steady rate. However, assuming that the rate of increase is constant allows you to estimate future data.

Problem-Solving Tip

Tools When graphing by hand, using graph paper and a straightedge can help you make your graphs more accurate.

(continued on the next page)

Source: IMF, CEIC


Sample answer: I need to graph a system of equations that represents this situation and find the intersection. How can I write the system of equations that I need to graph? I can review finding the average rate of change and writing equations in slope-intercept form.

Sample answer: I will find the average rate of change for both countries. Then I will write a system of equations to represent the situation. I will graph the system and find the intersection. I will use what I have learned about graphing equations to help me graph the system.

y = 1 ____ 150 x + 1.38

y = 1 ___ 60 x + 1.29

1 ____ 150

1 ___ 60


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Study Tip

Labeling Axes It is important to clearly define and label axes in a real-world situation. The intersection does not mean that the population will be the same in year 9. Since the x-axis represents the number of years after 2016, you must add the x-coordinate of the intersection to 2016 to find the year.

Think About It!

Can a value of x that is not a whole number be a viable solution? Justify your argument.

Sample answer: Yes; India and China could reach the same population partway through a year.

Graph the system of equations.

Populations

India

China

4 8 12 16 20Po

pula

tion

(bill

ions

)1

1.2

1.4

1.6

1.8

0

Years After 2016

When will China and India have the same population?

4. How can you know that your solution is reasonable? Write About It! Write an argument that can be used to defend

your solution.

CheckOLYMPICS The number of men and women participating in the Winter Olympic Games has been steadily increasing in recent years. In the 19th Winter Olympics, 1389 men and 787 women participated. 1660 men and 1121 women participated in the 22nd Winter Olympics.Part A

Write and graph a system to describe the number of men and women participating if x represents the number of Winter Olympics after the 22nd Winter Olympics.

Part B

Use the graph to predict the Winter Olympics when the number of men and women participating will be the same.


Winter Olympics

Men

Women

8 16 24 32 40

Num

ber o

f Par

ticip

ants

1400

2200

3000

3800

4600

0

Number of Winter OlympicsAfter 22nd


y = 271 ___ 3 x + 1660

y = 334 ____ 3 x + 1121

Sample answer: 47th Winter Olympics

2025

From the graph, I estimated that the solution was 9 years after 2016, or in 2025. My answer is reasonable because when I substitute 9 in for x in each equation, the value of y is 1.44


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Learn Using Systems to Solve Linear EquationsKey Concept • Using Systems to Solve Linear EquationsStep 1 Write a system by setting each expression equal to y.Step 2 Graph the system.Step 3 Find the intersection.

Example 8 Use a System to Solve a Linear EquationUse a system of equations to solve −6x + 8 = −4.

Step 1 Write a system.

Write a system of equations. Set each side of −6x + 8 = −4 equal to y.

Step 2 Graph the system.

Enter the equations and graph.

Step 3 Find the intersection.

Step 4 Check your solution.

−6x + 8 = −4 Original equation

−6( ) + 8 = −4 Substitution.

+ 8 = −4 Multiply.

= −4 Add.

CheckUse a system of equations and your graphing calculator to solve −3.2x − 5.8 = 2.8x + 7. Round to the nearest hundredth, if necessary.

x =


Learn Solving Systems of Equations by Using Graphing TechnologyYou can use a graphing calculator to graph and solve a system of equations by following these steps.

Step 1 Isolate y in each equation.

Step 2 Graph the system.

Step 3 Find the intersection.

Talk About It!

How do you know that the point of intersection satisfies both equations?

Go Online to see how to use a graphing calculator with this example.

Go Online You can watch a video to see how to graph systems of equations on a graphing calculator.


y = −6x + 8

y = −4

(2, −4)

−2−12

−4

−2.13

Sample answer: Every point on the line of an equation is a solution of that equation. Since the point of intersection lies on both lines, it is a solution of both equations.


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Example 9 Solve a System of EquationsSolve the system of equations.−1.38x − y = 5.130.62x + 2y = 1.60

Step 1 Isolate y.

Solve each equation for y.

−1.38x − y = 5.13 First equation −y = + x Add 1.38x to each side. y = − x Multiply each side by −1.

0.62x + 2y = 1.60 Second equation 2y = − Subtract 0.62x from each side. y = x − Divide each side by 2.

Step 2 Graph the system.Enter the equations and graph.

Step 3 Find the intersection.The solution is approximately .

CheckWhat is the solution to the system of equations?

2.29x − 4.41y = 6.524.16x + 1.11y = 4.72( , )

Example 10 Write and Solve a System of EquationsBUSINESS Denzel is starting a food truck business to sell gourmet grilled cheese sandwiches. He has spent $34,000 on the truck, equipment, permits, and other start-up costs. Each sandwich costs about $1.32 to make, and he sells them for $7.

How many sandwiches does Denzel need to sell to start earning a profit?

Let x = the number of . Let = total cost or revenue.

Total Cost: y = 1.32x Total Revenue: y =

Step 1 Graph the system.Enter the equations and graph.

Step 2 Find the intersection.The solution is approximately ( , ).This means that after Denzel has sold sandwiches, he will begin to earn a .





5.13 1.38−5.13 1.38

1.60 0.62x0.80 0.31x

−1.21 −2.11

grilled cheese sandwiches soldy

+ 34,0007x

(−5.54, 2.52)

5985.92 41,901.415986

profit


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Graphing Systems of Equations - MR. JONES · 2020. 1. 7. · Learn Solving Systems of Equations by Graphing You can solve a system of equations by graphing each equation carefully