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How can you find the solution of a system of linear equations by graphing?ESSENTIAL QUESTION

Solving a Linear System by GraphingA system of linear equations, also called a linear system, consists of two or more linear equations that have the same variables. A solution of a system of linear equations with two variables is an ordered pair that satisfies all of the equations in the system. The values of the variables in the ordered pair make each equation in the system true.

Systems of linear equations can be solved by graphing and by using algebraic methods. In this lesson you will learn to solve linear systems by graphing the equations in the system and analyzing how those graphs are related.

Solve the system of linear equations below by graphing. Check your answer.

{ -x + y = 3 2x + y = 6

STEP 1 Find the intercepts for each equation, plus a third point for a check.Graph each line.

-x + y = 3 2x + y = 6

x-intercept: −3 x-intercept: 3

y-intercept: 3 y-intercept: 6

third point: (3, 6) third point: (-1, 8)

STEP 2 Find the point of intersection.

The two lines appear to intersect at (1, 4).

STEP 3 Check to see if (1, 4) makes both equations true.

-x + y = 3 2x + y = 6

-(1) + 4 =? 3 2(1) + 4 =? 6

3 = 3✓ 6 = 6✓

The solution is (1, 4).

Solve the system of linear equations below by graphing. Check your answer.

EXAMPLE 1

L E S S O N

9.1Solving Linear Systems by Graphing

A-REI.3.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Also A-CED.1.3

A-REI.3.6

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1. Solve the system of linear equations below by graphing. Check your answer.

{ x + y = 8 x - y = 2

Solution:

YOUR TURN

Special Systems of Linear EquationsWhen a system of equations consists of two lines that intersect in one point, as shown in Figure 1 below, there is exactly one solution to the system.

If two linear equations in a system have the same graph, as shown in Figure 2, the graphs are coincident lines, or the same line. There are infinitely many solutions of the system because every point on the line represents a solution of both equations.

A system with at least one solution is a consistent system. Consistent systems can either be independent or dependent.

• An independent system has exactly one solution. The graph of an independent system consists of two intersecting lines.

• A dependent system has infinitely many solutions. The graph of a dependent system consists of two coincident lines.

When the two lines in a system do not intersect, as shown in Figure 3, they are parallel lines. There are no ordered pairs that satisfy both equations, so there is no solution. A system that has no solution is an inconsistent system.

The table below summarizes how systems of linear equations can be classified.

Classification of Systems of Linear Equations

Classification Consistent and Independent

Consistent and Dependent Inconsistent

Number of Solutions Exactly one Infinitely many None

Description Different slopes Same slope, same y-intercept

Same slope,differenty-intercepts

Graph

Figure 1 Figure 2 Figure 3

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3x - y = 1

x + y = 7x + y = 32x + 2y = 6

Graph each system of linear equations. Classify each system.

3. { 2x + 2y = 8 x - y = 4 4. { y = 2x - 4 y = 2x + 6

YOUR TURN

Use the graph to solve each system of linear equations. Classify each system.

A { x + y = 7 2x + 2y = 6

The lines do not intersect and appear to be parallel lines.

This system has no solution. The system is inconsistent.

B { 2x + 2y = 6 x + y = 3

The equations have the same graph, so the graphs are coincident lines.

This system has infinitely many solutions. The system is consistent and dependent.

REFLECT

2. Use the graph above to identify two lines that represent a linear system with exactly one solution. What are the equations of the lines? Explain your reasoning.

Use the graph to solve each system of linear equations. Classify each system.

EXAMPLE 2

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Why is the greatest possible number of solutions one

for a system of linear equations whose graph

shows two distinct lines?

A-REI.3.6

Math TalkMathematical Practices

Why is the greatest possible

Math Talk

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5. Estimate the solution for the linear system by graphing.

{ x - y = 3 x + 2y = 4

Approximate solution:

YOUR TURN

Estimating a Solution by GraphingYou can estimate a solution for a linear system by graphing and then check your estimate to determine if it is an approximate solution.

Estimate the solution for the linear system by graphing.

{ x + 2y = 2 2x - 3y = 12

STEP 1 Graph each equation by finding intercepts.

x + 2y = 2 2x - 3y = 12

x-intercept: 2 x-intercept: 6y-intercept: 1 y-intercept: -4

STEP 2 Find the point of intersection.

The two lines appear to intersect at about ( 4 1 _ 4 , -1 1 _ 8 ) .

STEP 3 Check if ( 4 1 _ 4 , -1 1 _ 8 ) is an approximate solution.

x + 2y = 2 2x - 3y = 12

4 1 _ 4 + 2 (-1 1 _ 8 ) =? 2 2(4 1 _ 4 ) - 3 (-1 1 _ 8 ) =? 12

4 1 _ 4 + (-2 1 _ 4 ) =? 2 8 1 _ 2 - (-3 3 _ 8 ) =? 12 2 = 2✓ 11 7 _ 8 ≈ 12✓

The point ( 4 1 _ 4 , -1 1 _ 8 ) does not make both equations true, but it is acceptable since 11 7 _ 8 is close to 12. So, ( 4 1 _ 4 , -1 1 _ 8 ) is an approximate solution.

EXAMPLE 3 A-REI.3.6

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x + y = -12x + 2y = -2

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Solve the system of linear equations by graphing. Check your answer. (Example 1)

1. { x - y = 7 2x + y = 2

x - y = 7 2x + y = 2

x-intercept: x-intercept:

y-intercept: y-intercept:

The solution of the system is .

Does your solution check in both original equations?

Use the graph to solve and classify each system of linear equations. (Example 2)

2. { x + y = -1 -2x + 2y = -2

Is the system consistent?

Is the system independent?

Solution:

3. { -x + y = 3 -2x + 2y = -2

Is the system consistent?

Solution:

Estimate the solution for the system of linear equations by graphing. (Example 3)

4. { x + y = -1 2x - y = 5

x + y = -1 2x - y = 5

x-intercept: x-intercept:

y-intercept: y-intercept:

The two lines appear to intersect at

.

5. How does graphing help you solve a system of linear equations?

ESSENTIAL QUESTION CHECK-IN???

Guided Practice

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9.1 Independent Practice

Name Class Date

Solve each system of linear equations by graphing. Check your answer.

6. { x - y = -2 2x + y = 8

Solution:

7. { x - y = -5 2x + 4y = -4

Solution:

8. { x + 2y = -8 -2x - 4y = 4

Solution:

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Estimate the solution for the linear system by graphing.

9. { x + y = 5 x - 3y = 3

Approximate solution:

Graph each system. Then classify it as consistent and independent, consistent and dependent, or inconsistent.

10. { x + 2y = 6 x = 2

11. { 2x - y = -6 4x - 2y = -12

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12. Use the graphed system below to find an approximate solution. Then write a system of equations that represents the system.

13. Communicate Mathematical Ideas When a system of linear equations is graphed, how is the graph of each equation related to the solutions of that equation?

14. Critical Thinking Write sometimes, always, or never to complete the following statement:

If the equations in a system of linear equations have the same slope, there are infinitely many solutions for the system.

15. Suppose you use the graph of a system of linear equations to estimate the solution. Explain how you would check your estimate to determine if it is an approximate solution.

16. Without graphing, describe the graph of this linear system of equations.

{ x = 3 y = 4

What is the solution of the linear system?

17. How can you recognize a dependent system of equations by analyzing the equations in the

system?

18. How would you classify a system of equations whose graph is composed of two lines with different slopes and the same y-intercepts? What is the solution to the system?

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19. Sophie and Marcos are each saving for new bicycles. So far, Sophie has $10 saved and can earn $5 per hour walking dogs. Marcos has $4 saved and can earn $8 per hour at his family’s plant nursery. After how many hours of work will Sophie and Marcos have saved the same amount? What will that amount be? Use the system of equations below to complete the graph.

Sophie: y = 5x + 10Marcos: y = 8x + 4

20. Represent Real-World Problems Cora ran 1 mile last week and will run

7 miles per week from now on. Hana ran 2 miles last week and will run 4 miles

per week from now on. The system of linear equations { y = 7x + 1 y = 4x + 2 can be

used to represent this situation. Explain what x and y represent in the equations.

21. Explain the Error Jake was asked to give an example of an inconsistent system of linear equations. He wrote the system shown below. Explain Jake’s error.

{ y = 2x - 4 y = x - 4

22. Draw Conclusions The equations in a system of linear equations have different slopes and the same y-intercept. Can you find the solution without graphing? Explain.

FOCUS ON HIGHER ORDER THINKING Work Area

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