CorrectionKey=NL-B;CA-B Name Class Date 11.1 Solving Linear … · 2016. 2. 19. · Explain 1 Solving Consistent, Independent Linear Systems by Graphing A consistent system is a system

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Explore Types of Systems of Linear EquationsA 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 any ordered pair that satisfies all of the equations in the system.

A Describe the relationship between the two lines in Graph A.

B What do you know about every point on the graph on a linear equation?

C How many solutions does a system of two equations have if the graphs of the two equations intersect at exactly one point?

D Describe the relationship between the two lines that coincide in Graph B.

E How many solutions does a system of two equations have if the graphs of the two equations intersect at infinitely many points?

Graph A

Graph B

Module 11 479 Lesson 1

11.1 Solving Linear Systems by Graphing

Essential Question: How can you find the solution of a system of linear equations by graphing?

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Describe the relationship between the two lines in Graph C.

How many solutions does a system of two equations have if the graphs of the two equations do not intersect?

Reflect

1. Discussion Explain why the solution of a system of two equations is represented by any point where the two graphs intersect.

Explain 1 Solving Consistent, Independent Linear Systems by Graphing

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

An independent system has exactly one solution. The graph of an independent system consists of two lines that intersect at exactly one point. A dependent system has infinitely many solutions. The graph of a dependent system consists of two coincident lines, or the same line.

A system that has no solution is an inconsistent system.

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

A ⎧

⎨

⎩ 2x + y = 6

-x + y = 3

Find the intercepts for each equation, plus a third point for a check. Then graph.

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

x-intercept: 3 x-intercept: –3

y-intercept: 6 y-intercept: 3

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

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

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

2 (1) + 4 ≟ 6 - (1) + 4 ≟ 3

6 = 6 3 = 3

The point satisfies both equations, so the solution is (1, 4) .

Graph C

4

6

8

-2 42

y

x

2x + y = 6

-x + y = 3

2

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B ⎧

⎨

⎩ y = 2x - 2

3y + 6x = 18

Find the intercepts for each equation, plus a third point for a check. Then graph.

y = 2x - 2 3y + 6x = 18

x-intercept: x-intercept:

y-intercept: y-intercept:

third point: (3, ) third point: (1, ) The two lines appear to intersect at . Check.

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

≟ 2 ( ) - 2 + 2 ( ) ≟ 6

= = 6

The point satisfies both equations, so the solution is .

Reflect

2. How do you know that the systems of equations are consistent? How do you know that they are independent?

Your Turn

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

3. ⎧

⎨

⎩ y = −2x − 2

x + 2y = 2

4. ⎧

⎨

⎩ y = 2x + 8

−x + y = 6

4

6

-2-2 4 62

y

x

2

0

2

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6

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Explain 2 Solving Special Linear Systems by Graphing

Example 2 Solve the special system of equations by graphing and identify the system.

A ⎧

⎨

⎩ y = 2x - 2

-2x + y = 4

Find the intercepts for each equation, plus a third point for a check.

y = 2x - 2 -2x + y = 4

x-intercept: 1 x-intercept: -2

y-intercept: -2 y-intercept: 4

third point: (2, 2) third point: (2, 8)

The two lines don’t intersect, so there is no solution. The two lines have the same slope and different y-intercepts so they will never intersect. This is an inconsistent system.

B ⎧

⎨

⎩ y = 3x - 3

-3x + y = -3

Find the intercepts for each equation, plus a third point for a check.

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

x-intercept: x-intercept:

y-intercept: y-intercept:

third point: (2, ) third point: (2, ) The two lines coincide, so there are solutions.

They have the same slope and y-intercept; therefore, they are line(s) / equation(s).

This is a and system.

\ Your Turn

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

5. ⎧ ⎨

⎩ y = -x - 2

x + y + 2 = 0

6. ⎧

⎨

⎩ y = 2 _ 3 x - 1

- 2 _ 3 x + y = 1

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Explain 3 Estimating Solutions of Linear Systems by GraphingYou can estimate the solution of a linear system of equations by graphing the system and finding the approximate coordinates of the intersection point.

Example 3 Estimate the solution of the linear system by graphing.

A ⎧ ⎨

⎩ x - 3y = 3

-5x + 2y = 10

Graph the equations using a graphing calculator.

Y1 = (3 – X)/(–3) and Y2 = (10 + 5X)/2

Find the point of intersection.

The two lines appear to intersect at about (−2.8, −1.9) .

B ⎧

⎨

⎩ 6 - 2y = 3x

y = 4x + 8

Graph each equation by finding intercepts.

The two lines appear to intersect at about .

Check to see if makes both equations true.

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

6 - 2 ( ) ≟ 3 ( ) ≟ 4 ( ) + 8

≈ ≈

The point does not satisfy both equations, but the results are close.

So, is an approximate solution.

Your Turn

Estimate the solution of the linear system of equations by graphing.

7.

⎧ ⎨

⎩ 2y = -5x + 10

-15 = -3x + 5y

8. ⎧

⎨

⎩ 3x + 3y = -9

y = 1 _ 2 x - 1

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2

y

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x

-2-4

-2

4

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-2-4

-2

2

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Explain 4 Interpreting Graphs of Linear Systems to Solve Problems

You can solve problems with real-world context by graphing the equations that model the problem and finding a common point.

Example 4 Rock and Bowl charges $2.75 per game plus $3 for shoe rental. Super Bowling charges $2.25 per game and $3.50 for shoe rental. For how many games will the cost to bowl be approximately the same at both places? What is that cost?

Analyze Information

Identify the important information.

• Rock and Bowl charges $ per game plus $ for shoe rental.

• Super Bowling charges $ per game and $ for shoe rental.

• The answer is the number of games played for which the total cost is approximately the same at both bowling alleys.

Formulate a Plan

Write a system of linear equations, where each equation represents the price at each bowling alley.

Solve

Graph y = 2.75x + 3 and y = 2.25x + 3.50.

The lines appear to intersect at . So, the cost at both

places will be the same for game(s) bowled and that

cost will be .

Justify and Evaluate

Check using both equations.

2.75 ( ) + 3 = 2.25 ( ) + 3.5 =

Reflect

9. Which bowling alley costs more if you bowl more than 1 game? Explain how you can tell by looking at the graph.

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Games

Cost

($)

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Your Turn

10. Video club A charges $10 for membership and $4 per movie rental. Video club B charges $15 for membership and $3 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost? Write a system and solve by graphing.

Elaborate

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

12. Essential Question Check-In How does graphing help you solve a system of linear equations?

1. Is the following statement correct? Explain.

A system of two equations has no solution if the graphs of the two equations are coincident lines.

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

2. ⎧

⎨

⎩ y = 2x - 4

x + 2y = 12

3.

⎧

⎨

⎩

y = - 1 _ 3 x + 2

y + 4 = - 4 _ 3 x

Evaluate: Homework and Practice

• Online Homework• Hints and Help• Extra Practice

0

10

20

30

40

2 4 6 8

x

Number of movies

Cost

($)

y

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-4-8 84

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4. ⎧

⎨

⎩ y = -x - 6

y = x 5. ⎧

⎨

⎩ y = 4 _ 3 x - 4

y = 4

6. ⎧

⎨

⎩ y = -2x + 2

y + 2 = 2x

7.

⎧

⎨

⎩

y = 1 _ 2 x + 5

2 _ 3 x + y = -2

8.

⎧

⎨

⎩

y = 1 _ 2 x - 2

- 1 _ 2 x + y - 3 = 0

9. ⎧

⎨

⎩ y = 2x + 4

-4x + 2y = 8

10. ⎧

⎨

⎩ y = -3x + 1

12x + 4y = 4

11. ⎧

⎨

⎩ y = 4x + 4

-4x + y + 4 = 0

0

8

4

-4-8 84

y

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-4

-8

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-4-8 84

y

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-8

-4

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-4

-8

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-4-8 84

y

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-4

-8

4

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-4-8 4 8

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-4

-8

8

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-4-8 84

y

x0

-4

-8

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-8 -4 84

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y12.

⎧

⎨

⎩

y = - 3 _ 4 x + 1 _ 4

3 _ 4 x + y - 2 = 0

13. ⎧

⎨

⎩ y = 5x - 1

-5x + y + 4 = 3

Estimate the solution of the linear system of equations by graphing.

14. ⎧

⎨

⎩ 3y = -5x + 15

-14 = -2x + 7y

15. ⎧

⎨

⎩ 2y + 5x = 14

-35 = -5x + 7y

16. ⎧

⎨

⎩ 4 _ 7 y = x + 4

2x - 5y = 10

17. ⎧

⎨

⎩ 6y = 5x + 30

2 = - 2 _ 7 x + y

0

-4

-8

8

4

-8 -4 84

y

x0

-4

-8

8

4

-8 -4 84

y

x

0

-4

-8

8

4

-8 -4 84

y

x0

-4

-8

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-8 -4 84

y

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-8 -4 84

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Solve by graphing. Give an approximate solution if necessary.

18. Wren and Jenni are reading the same book. Wren is on page 12 and reads 3 pages every night. Jenni is on page 7 and reads 4 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?

19. Rusty burns 6 calories per minute swimming and 10 calories per minute jogging. In the morning, Rusty burns 175 calories walking and swims for x minutes. In the afternoon, Rusty will jog for x minutes. How many minutes must he jog to burn at least as many calories y in the afternoon as he did in the morning? Round your answer up to the next whole number of minutes.

20. A gym membership at one gym costs $10 every month plus a one-time membership fee of $15, and a gym membership at another gym costs $4 every month plus a one-time $40 membership fee. After about how many months will the gym memberships cost the same amount?

21. Malory is putting money in two savings accounts. Account A started with $150 and Account B started with $300. Malory deposits $16 in Account A and $12 in Account B each month. In how many months will Account A have a balance at least as great as Account B? What will that balance be?



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22. Critical Thinking Write sometimes, always, or never to complete the following statements.

a. If the equations in a system of linear equations have the same slope,

there are infinitely many solutions for the system.

b. If the equations in a system of linear equations have different slopes, there

is one solution for the system.

c. If the equations in a system of linear equations have the same slope and a different

y-intercept, there is any solution for the system.

H.O.T. Focus on Higher Order Thinking

23. Critique Reasoning Brad classifies the system below as inconsistent because the equations have the same y-intercept. What is his error?

⎧

⎨

⎩ y = 2x - 4

y = x - 4

24. Explain the Error Alexa solved the system

⎧

⎨

⎩ 5x + 2y = 6

x - 3y = -4

by graphing and estimated the solution to be about (1.5, 0.6) . What is her error? What is the correct answer?

25. Represent Real-World Problems Cora ran 3 miles last week and will run 7 miles per week from now on. Hana ran 9 miles last week and will run 4 miles per week

from now on. The system of linear equations ⎧

⎨

⎩ y = 7x + 3

y = 4x + 9

can be used to represent

this situation. Explain what x and y represent in the equations. After how many weeks will Cora and Hana have run the same number of miles? How many miles? Solve by graphing.



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Lesson Performance TaskA boat takes 7.5 hours to make a 60-mile trip upstream and 6 hours on the 60-mile return trip. Let v be the speed of the boat in still water and c be the speed of the current. The upstream speed of the boat is v - c and the downstream boat speed is v + c.

a. Use the distance formula to write a system of equations relating boat speed and time to distance, one equation for the upstream part of the trip and one for the downstream part.

b. Graph the system to find the speed of the boat in still water and the speed of the current.

c. How long would it take the boat to travel the 60 miles if there were no current?

v

c

0

2

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CorrectionKey=NL-B;CA-B Name Class Date 11.1 Solving Linear … · 2016. 2. 19. · Explain 1 Solving Consistent, Independent Linear Systems by Graphing A consistent system is a system