ch3 ans

Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

1

Name Class Date

3-1 ELL Support Solving Systems Using Tables and Graphs

Use the chart below to review vocabulary. Th ese vocabulary words will help you complete this page.

Use the vocabulary above to fi ll in the blanks.

1. On the test tomorrow, you will linear systems according to how many solutions each system has.

2. Linear systems are in order to give us information about them.

3. Th e to a system of equations is a point of intersection on their graphs.

4. You can a system of equations using a table or a graph.

Example Linear systems are classifi ed as consistent or inconsistent.

1 solution consistent no solution inconsistent

Solve each system of equations and classify the system as consistent or inconsistent.

5. e y 5 2x 2 7y 5 2x 1 9

6. e 3x 2 y 5 6x 1 y 5 2

Examples Word Explanations Classify (verb) KLAS uh fahy

To arrange into groups thathave similar characteristics

We classify linear systems based upon how many solutionsthe system has.

Classified (verb) KLAS uh fahyd

Arranged into groups Linear systems are classified asconsistent or inconsistent.

Solution (noun) suh Loo shun

The ordered pair that makesboth equations true

(2, 24) is the solution to thesystem of equations3x 1 y 5 2 and x 2 2y 5 10.

Solve (verb) sahlv

To work out a correct answerto the problem

Solve the system of equations using a table.

classify

classifi ed

solution

solve

(2, 0); consistentno solution; inconsistent


2

Name Class Date

3-1 Think About a Plan Solving Systems Using Tables and Graphs

Sports You can choose between two tennis courts at two university campuses to learn how to play tennis. One campus charges $25 per hour. Th e other campus charges $20 per hour plus a one-time registration fee of $10.

a. Write a system of equations to represent the cost c for h hours of court use at each campus.

b. Graphing Calculator Find the number of hours for which the costs are the same.

c. Reasoning If you want to practice for a total of 10 hours, which university campus should you choose? Explain.

1. What is an equation that represents the cost c for h hours of court use for the fi rst campus?

2. What is an equation that represents the cost c for h hours of court use for the second campus?

3. What is one method you can use to fi nd the number of hours for which the costs are the same?

4. What is another method you can use to fi nd the number of hours for which the costs are the same?

5. Use one of your methods to fi nd the number of hours for which the costs are the same.

6. What happens to the cost at the two campuses after you have practiced for the number of hours you found in Exercise 5?

7. If you want to practice for a total of 10 hours, which university campus should you choose? Explain.

c 5 25h

2 h

c 5 20h 1 10

Answers may vary. Sample: Graph both equations and fi nd the point of intersection.

Answers may vary. Sample: Use a table to list x-values until the corresponding y-values

The cost at the fi rst campus increases by $25 per hour, while the cost at the second

The second campus; after 2 hours, the cost of the fi rst campus will always be greater

match each other.

campus increases by $20 per hour.

than the cost of the second campus, because the slope of the equation for the fi rst

campus is greater than the slope of the equation for the second campus.

Prentice Hall Gold Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

Name Class Date

Solve each system by graphing or using a table. Check your answers.

1. e y 5 x 2 2x 1 y 5 10

2. e y 5 7 2 x

x 1 3y 5 11 3. e x 2 2y 5 10

y 5 x 2 11

4. e 5x 1 y 5 115x 2 y 5 1

5. e x 1 y 5 21x 2 y 5 3

6. e 2x 2 2y 5 212x 1 2y 5 10

7. e 4x 1 3y 5 2162x 1 y 5 4

8. e y 5 23x

x 1 y 5 2 9. e y 5

23x 2 5

y 5 223x 2 3

10. e y 512x 1 3

y 5 214x 2 3

11. e 2x 2 4y 5 243x 2 4y 5 4

12. e x 1 y 5 6x 2 y 5 4

Write and solve a system of equations for each situation. Check your answers.

13. Your school sells tickets for its winter concert. Student tickets are $5 and adult tickets are $10. If your school sells 85 tickets and makes $600, how many of each ticket did they sell?

14. A grocery store has small bags of apples for $5 and large bags of apples for $8. If you buy 6 bags and spend $45, how many of each size bag did you buy?

15. Th e spreadsheet below shows the monthly income and expenses for a new business.

a. Use your graphing calculator to fi nd linear models for income and expenses as functions of the number of the month.

b. In what month will income equal expenses?

3-1 Practice Form G

Solving Systems Using Tables and Graphs

A

Month

1 May

2 June

3 July

4 August

B

Income

$1500

$3500

$5500

$7500

C

Expenses

$21,400

$18,800

$16,200

$13,600

(6, 4)

(2, 1)

(24, 0) Q32, 24R

(5, 2)

(1, 22)

(21, 3)

(12, 1)

(2, 3)

(28, 21) (2, 2) (5, 1)

5x 1 10y 5 600, x 1 y 5 85; 50 student tickets, 35 adult tickets

5x 1 8y 5 45, x 1 y 5 6; 1 small bag, 5 large bags

income: y 5 2000x 2 500; expenses: y 5 22600x 1 24,000

September


4

Name Class Date

Without graphing, classify each system as independent, dependent, or inconsistent.

16. e x 1 y 5 3y 5 2x 2 3

17. e 2x 1 y 5 3y 5 22x 2 1

18. e x 1 3y 5 922x 2 6y 5 218

19. e x 1 y 5 4y 5 2x 1 1

20. e x 1 3y 5 99y 1 3x 5 27

21. e 2x 1 2y 5 52x 1 3y 5 9

22. e 3x 1 2y 5 73x 2 15 5 26y

23. e 3x 1 3y 5 63x 1 3y 5 3

24. e x 1 y 5 11y 5 x 2 5

25. e x 1 2y 5 132y 5 7 2 x

26. e y 5 12 2 5xx 2 4y 5 26

27. e 25x 2 10y 5 02y 5 5x

28. You and your business partner are mailing advertising fl yers to your customers. You address 6 fl yers each minute and have already done 80. Your partner addresses 4 fl yers each minute and has already done 100. Graph and solve a system of equations to fi nd when the two of you will have addressed equal numbers of fl yers.

29. You are going on vacation and leaving your dog in a kennel. Kennel A charges $25 per day which includes a one-time grooming treatment. Kennel B charges $20 per day and a one-time fee of $30 for grooming.

a. Write a system of equations to represent the cost c for d days that your dog will stay at the kennel.

b. If your vacation is 7 days long, which kennel should you choose? Explain.

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.

30. one 31. none 32. an infi nite number

e y 5 5x 2 3?

e y 5 2x 1 3?

e y 5 3x 2 2?

33. Multiple Choice Which ordered pair of numbers is the solution of the system?

e x 1 y 5 23x 2 2y 5 0

(26, 23) (22, 21) (6, 23) (2, 1)

3-1 Practice (continued) Form G


20406080

100120140160

YouYour

Partner

2 4 6 8 10 12 14 16

x

y

O

independent

independent

independent

inconsistentindependent

independent

independent

dependent

inconsistent

inconsistent

dependent

dependent

y 5 6x 1 80, y 5 4x 1 100; 10 minutes

c 5 25d, c 5 20d 1 30

Kennel B; it costs $170, while Kennel A costs $175.

any equation with a different slope

any equation with the same slope but a different y-intercept

any equivalent equation

B

Name Class Date

Prentice Hall Foundations Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

5

3-1 Practice Form K



1. e y 5 2x 1 4y 5 25x 2 3

2. e x 2 y 5 2y 5 2x

3. e y 5 22x 2 22y 5 x 1 6

Write and solve a system of equations for each situation. Check your answers.

4. Each morning you do a combination of aerobics, which burns about 12 calories per minute, and stretching, which burns about 4 calories per minute. Your goal is to burn 416 calories during a 60-minute workout. How long should you spend on each type of exercise to burn the 416 calories?

5. Suppose 28 members of your class went on a rafting trip. Class members could either rent canoes for $16 each or rent kayaks for $19 each. Th e class spent a total $469. How many people rented canoes and how many people rented kayaks?

For Exercise 6, use your graphing calculator to fi nd a linear model for the set of data. In what year will the two quantities be equal?

6.

Year

Male Times (s)

Female Times (s)

Winning Times for the Olympic 100-Meter Butterfly

1972

54.27

63.34

SOURCE: www.infoplease.com

1976

54.35

60.13

1980

54.92

60.42

1984

53.08

59.26

1988

53.00

59.00

1992

53.32

58.62

1996

52.27

59.13

2000

52.00

56.61

2004

51.25

57.72

2008

50.58

56.73

(21, 2) (1, 21) (22, 2)

2156

x 1 y 5 60; 12x 1 4y 5 416; x 5 22 minutes aerobics; y 5 38 minutes stretching

x 1 y 5 28, 16x 1 19y 5 469; x 5 21 rented canoes, y 5 7 rented kayaks

Name Class Date


6

3-1 Practice (continued) Form K


Without graphing, does each system have zero, one, or infi nitely many solutions? To start, rewrite each equation in slope-intercept form.

7. e 4y 1 8 5 12x

y 2 5 5 3x 8. e 6y 2 3x 5 12

2y 5 x 1 4 9. e 1

5 y 5 x 215

x 5 11 2 y

e y 5 3x 2 2y 5 3x 1 5

Graph and solve each system.

10. e 2y 5 22x 1 92y 5 2x 1 6

11. e y 5 3x 2 8y 5 x 2 8

12. e y 1 5 5 4x2y 5 8x 1 2

13. Your business needs to ship a package to another store. Company A charges $2.50 per pound plus a $20 service charge. Company B charges $4.50 per pound without any service charge.

a. At what weight does it cost the same for both companies? b. If your package weighs 14 pounds, which shipping company should your

business use?

14. Reasoning Is it possible for a linear system with infi nitely many solutions to contain two lines with diff erent y-intercepts? No; a system with infi nitely manysolutions contains the same line.

O

y

x

�2

2

�4

2 4y � �2x � 9

6

y � � x � 312 O

y x

�6

�4

�8

2�2�4

y � x � 8

y � 3x � 8O

y

x

�6

�4

2�2�4

y � 4x � 5y � 4x � 1

zero solutions infi nitely many solutions one solution

(4, 1) (0, 28) no solution

10 pounds

Company A


7

Name Class Date

Multiple Choice

For Exercises 1–4, choose the correct letter.

1. Which system of equations is inconsistent?

e x 1 y 5 4x 2 y 5 3

e 6x 1 3y 5 122y 5 24x 1 4

e 2y 2 x 5 54y 5 2x 1 10

e23x 1 y 5 42y 5 26x 1 8

2. Which ordered pair of numbers is the solution of the system? e 2x 1 3y 5 122x 2 3y 5 4

(2, 3) (3, 2) (1, 22) (23, 6)

3. Which of the following graphs shows the solution of the system?

e 2x 1 2y 5 242x 2 2y 5 28

4. You and your friend are both knitting scarves for charity. You knit 8 rows each minute and already have knitted 10 rows. Your friend knits 5 rows each minute and has already knitted 19 rows. When will you both have knitted the same number of rows?

2.6 minutes 3 minutes 9.7 minutes 34 minutes

Short Response

5. Th e sides of an angle are two lines whose equations are 4x 1 y 5 12 and y 5 3x 2 2. An angle has its vertex at the point where the lines meet. Use a graph to determine the coordinates of the vertex. What are the coordinates of the vertex?

xO�4 �2

�2�4

2 4

24 y

xO�2

�2�4

2 4

24 y

xO 2 4�4 �2

�2

2

4

�4

y

xO 2 4�2

�2

2

4

�4

y

3-1 Standardized Test Prep Solving Systems Using Tables and Graphs

y

xO 4

�4�4�8 8

4

8

12

C

G

D

G

[2] graph at right; (2, 4)[1] correct coordinates, without graph[0] no answer given


8

Name Class Date

When navigating their crafts, ship captains might be seen drawing lines on a large map using a straightedge. Th is procedure is used to plot intercept courses, which help to predict where the craft might meet an oncoming plane or ship.

Air traffi c controllers must also plot these intercept courses so that an oncoming plane does not meet another plane. To determine whether two ships or planes are on a collision course, it is important to know along which lines they are moving, where they are at a given moment, and how fast they are traveling.

Solve. You might wish to use a drawing to help you solve the problems.

1. Plane A is 40 mi south and 100 mi east of Plane B. Plane A is fl ying 2 mi west for every mile it fl ies north, while Plane B is fl ying 3 mi east for every mile it fl ies south.

a. Where do their paths cross? b. Which plane must fl y farther? c. What ratio of the speed of Plane B to the speed of Plane A would produce a

midair collision?

2. A commercial freight carrier is fl ying at a constant speed of 400 mi/h and is traveling 4 mi east for every 3 mi north. A private plane is observed to be 210 mi due east of the commercial carrier, traveling 12 mi north for every 5 mi west.

a. If it is known that the two planes are on a collision course, how fast is the private plane fl ying?

b. When will the collision take place if it is not averted?

3. A cruise ship is traveling in the Atlantic Ocean at a constant rate of 40 mi/h and is traveling 2 mi east for every 5 mi north. An oil tanker is 350 mi due north of the cruise ship and is traveling 1 mi east for every 1 mi south.

a. How far is each ship from the point at which their paths cross? b. What rate of speed for the oil tanker would put it on a collision course with

the cruise ship?

3-1 Enrichment Solving Systems Using Tables and Graphs

20 mi south and 60 mi east of Plane B’s current locationPlane B

"2 : 1 or "21

260 mi/h in 12 hour

cruise ship: about 269.3 mi; oil tanker: about 141.4 mi

about 21 mi/h


9

Name Class Date

As you solve a system of equations, remember the following ideas.

• Lines that have the same slopes but diff erent y-intercepts are parallel and will never intersect. Th ese systems are inconsistent.

• Lines that have both the same slope and the same y-intercept are the same line and will intersect at every point. Th ese systems are dependent.

• Lines that have diff erent slopes will intersect, and the system will have one solution. Th ese systems are independent.

Problem

Using a graph or a table, what is the solution of the system of equations? e 2x 1 y 5 8y 2 x 5 2

y 5 22x 1 8 Write both equations in y 5 mx 1 b form.

y 5 x 1 2

Graph the line y 5 22x 1 8. Graph the line y 5 x 1 2. Circle the point of intersection.

x 5 2, y 5 4 Determine the x- and y -coordinates of the point of intersection.

Th e solution is the ordered pair (2, 4).

Check 2(2) 1 4 0 8 Check by substituting the solution into both equations.

4 1 4 0 8 8 5 8 �

4 2 2 0 22 5 2 �

Exercises


1. e 3x 1 y 5 63x 1 y 5 3

2. e22x 1 y 1 3 5 0x 2 1 5 y

3. e x 1 y 5 3y 5 3x 2 1

4. e y 5 1 2 x

2x 1 y 5 4 5. e2x 1 2y 5 2

3x 1 2y 5 26 6. e2x 1 y 5 22

22x 1 3y 5 23

7. Which point lies on both Line 1 and Line 2?

(0, 0) (1.875, 1.875)

(2.05, 2.05) (2, 2)

y � �2x � 8

y � x � 2 x

y

O2 4 6 8�2

468

3-1 Reteaching Solving Systems Using Tables and Graphs

xO 2Line 1

4

Line 24

�4

y

(1, 3)

(3, 22)

(2, 1)

(22, 0)

C

(1, 2)

(3, 1)


10

Name Class Date

Problem

Th e table shows the winning times for the Olympic 400-M dash. Use your graphing calculator to fi nd linear models for women’s and men’s winning times. Assuming the trends in the table continue, when will the women’s winning time and the men’s winning time be equal? What will that winning time be?

Step 1 Enter the data into lists on your calculator.

L1: number of years since 1968 (value for x)

L2: men’s winning times in seconds (value for y1)

L3: women’s winning times in seconds (value for y2)

Step 2 Use LinReg(ax 1 b) to fi nd linear models. Th is determines the equation of the lines of best fi t for the selected data.

Use L1 and L2 for the men’s winning times.

Use L1 and L3 for the women’s winning times.

Step 3 Graph each model. Use the Intersect feature on the graphing calculator to fi nd the solution of the system. Th e solution is x 5 99.72093 and y 5 42.00168.

Th e linear model shows that if the table’s trends continue, the times for men and women will be equal about 100 years after 1968, in 2068. Th e winning time will be about 42 seconds.

Exercise 8. Th e table shows the winning times for Olympic 500-M speed skating.

Assuming these trends continue, when will the women’s winning time equal the men’s winning time? What will that winning time be?

Winning Times for the Olympic 400-M Dash (seconds)

Year 1968

52.03

43.86

51.08

1972

44.66

49.29

1976

44.26

1980

48.88

44.60

1984

48.83

44.27

1988

48.65

43.87

1992

48.83

43.50

1996

48.25

43.49

2000

49.11

43.84Men’sTimes

Women’sTimes

SOURCE: International Olympic Committee

�

��

Intersectionx � 99.72093 y � 42.00168

Winning Times for the Olympic 500-M Speed Skating (seconds)

Year 1968

46.10

40.30

43.33

1972

39.44

42.76

1976

39.17

1980

41.78

38.03

1984

41.02

38.19

1988

39.10

36.45

1992

40.33

37.14

1994

39.25

36.33

1998

38.21

35.59Men’sTimes

Women’sTimes

SOURCE: International Olympic Committee

3-1 Reteaching (continued)


2028; 31.265 seconds


11

Name Class Date

3-2 ELL Support Solving Systems Algebraically

Th e column on the left shows the steps used to solve a problem using a system of equations. Use the column on the left to answer each question in the column on the right.

Problem Solve by Setting up and Solving a System of Equations.

A girl received $100 for her birthday. At the store she can buy 1 video game and 3 DVDs for $100. She could also buy 2 video games and 1 DVD for $100. Write and solve a system of equations to determine the cost of a video game and the cost of a DVD.

1. Read the title of the Problem. What process are you going use to solve the problem?

Relate 2. Why is the cost of one DVD multiplied by 3?

Defi ne

Let v 5 cost of one video game and d 5 cost of one DVD.

3. Explain why the cost of a video game and the cost of a DVD are represented by variables.

Write 4. Why was the equation solved for v rather than d?

Check

Substitute d 5 20 and v 5 40 into the original equations.

5. Explain why the solution d 5 20 and v 5 40 should be checked in both equations.

Choose one equation and solve for v.

Write the other equation.

Substitute the expression for v into the other equation. Solve.

Substitute the value for d into one of the equations. Solve for v.

v 1 3d 5 100 v 5 100 2 3d

2v 1 d 5 100

2(10023d) 1 d 5 100

d 5 20

v 1 3(20) 5 100

v 5 40

acost of oneb 5 100video game DVDcost of one

1 3

2 acost of oneb 1 acost of oneb 5 100video game DVD

40 1 3(20) 5 100 2(40) 1 20 5 10040 1 60 5 100 80 1 20 5 100

100 5 100 100 5 100

Answers may vary. Sample:

It must be a solution to both

equations.


Write and solve a system of

equations.


There are three DVDs.


These are unknowns.


Solving for d would involve

fractions.


12

Name Class Date

3-2 Think About a PlanSolving Systems Algebraically

Chemistry A scientist wants to make 6 milliliters of a 30% sulfuric acid solution. Th e solution is to be made from a combination of a 20% sulfuric acid solution and a 50% sulfuric acid solution. How many milliliters of each solution must be combined to make the 30% solution?

Know

1. Th e scientist will begin with z z% and z z% solutions.

2. Th e scientist wants to make z zml of 30% solution.

Need

3. To solve the problem you need to defi ne:

Plan

4. What are two equations you can write to model the situation?

5. Which method should you use to solve the system of equations? Explain.

6. Solve the system of equations.

7. How can you interpret the solutions in the context of the problem?

8. Do your solutions check? Explain.

20

x 5 ml of 20% solution

0.2x 1 0.5y 5 1.8 and x 1 y 5 6

Answers may vary. Sample: Substitution; it is easy to solve one of the

yes; 4 ml 1 2 ml 5 6 ml, and

y 5 ml of 50% solution

equations (x 1 y 5 6) and get one of the variables in terms of the other variable.

4 ml ? 0.2 1 2 ml ? 0.5 5 0.8 ml 1 1 ml 5 1.8 ml 5 6 ml ? 0.3

x 5 4; y 5 2

4 ml of the 20% solution and 2 ml of the 50% solution should be combined.

6

50


13

Name Class Date

3-2 Practice Form G

Solving Systems Algebraically

Solve each system by substitution. Check your answers.

1. e y 5 x 1 12x 1 y 5 7

2. e x 5 y 2 23x 2 y 5 6

3. e y 5 2x 1 35x 2 y 5 23

4. e 6x 2 3y 5 2332x 1 3y 5 21

5. e 2x 2 2y 5 73x 2 2y 5 10

6. e 4x 5 8y

2x 1 5y 5 27

7. e x 1 3y 5 24y 1 3x 5 0

8. e 3x 1 2y 5 93x 1 2y 5 3

9. e 2y 2 3x 5 4x 5 24

10. Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that day, you bought six oranges and three grapefruits for a total of $4.80. What is the price of each type of fruit?

Solve each system by elimination.

11. e x 1 y 5 10x 2 y 5 12

12. e2x 1 3y 5 212x 2 2y 5 22

13. e x 1 3y 5 17x 1 3y 5 11

14. e 4x 2 3y 5 224x 1 5y 5 14

15. e 3x 1 2y 5 103x 2 2y 5 19

16. e 2x 2 15y 5 114x 1 10y 5 18

17. e x 2 y 5 0x 1 y 5 2

18. e x 1 3y 5 24x 1 3y 5 20

19. e 3x 2 y 5 172x 1 y 5 18

20. Th ere are a total of 15 apartments in two buildings. Th e diff erence of two times the number of apartments in the fi rst building and three times the number of apartments in the second building is 5.

a. Write a system of equations to model the relationship between the number of apartments in the fi rst building f and the number of apartments in the second building s.

b. How many apartments are in each building?


21. e2x 1 y 5 35x 1 y 5 9

22. e25x 1 4y 5 225x 2 2y 5 4

23. e22x 1 y 5 2325x 2 y 5 23

24. e 14x 1 2y 5 1014x 2 5y 5 11

25. e 2x 1 15y 5 12x 1 10y 5 2

26. e 0.3x 1 0.4y 5 20.80.7x 2 0.8y 5 26.8

27. e 4x 1 3y 5 265x 2 6y 5 227

28. e 2x 1 2y 5 2104x 1 2y 5 211

29. e 1.2x 1 1.4y 5 2.70.4x 2 0.3y 5 0.9

(2, 3)

(23, 5)

(4, 6)

(4, 1)

(0, 3)

(6, 3)

1 orange costs $.50; 1 grapefruit costs $.60

f 1 s 5 15, 2f 2 3s 5 510 in the fi rst; 5 in the second

(2, 22) (3, 0) (24, 24)

(6, 4)

(1, 2)

(1, 1)

(1, 4)

(2, 22)

(22, 3)

(24, 5)(1, 22)

(23, 2) No solution

e (x, y): y 5 215x 1 1

5 f

(5, 22)

(0, 3)

(2.25, 0)

(4, 3) Q5, 215R

(4, 1) (5, 2)


14

Name Class Date

30. Writing Explain what it means when elimination results in an equation that is always true.

For each system, choose the solution method that seems easier to use. Explain why you made each choice. Solve each system.

31. eb 5 2a 2 5b 5 3 1 a

32. e 4x 2 2y 5 114x 1 3y 5 6

33. e 5p 1 2q 5 104p 1 2q 5 4

34. e j 2 3k 5 3 j 5 2k 1 15

35. Error Analysis You and your friend are solving the system 4x 2 y 5 5 and 4x 1 y 5 3. Your friend says there is no solution, and you say the solution is (1, 21). Who is correct? Explain.

36. You can buy DVDs at a local store for $15.49 each. You can buy them at an online store for $13.99 each plus $6 for shipping. How many DVDs can you buy for the same amount at the two stores?

37. Last year, a baseball team paid $20 per bat and $12 per glove, spending a total of $1040. Th is year, the prices went up to $25 per bat and $16 per glove. Th e team spent $1350 to purchase the same amount of equipment as last year. How many bats and gloves did the team purchase each year?

38. If the perimeter of the square at the right is 72 units, what are the values of x and y?

3y

2x



When elimination results in an equation that is always true, it means the equations in the system represent the same line.

You are correct; your friend mistakenly thought both x and y had opposite coeffi cients, causing both variables to “drop out,” giving a system with no solution.

4

22 bats, 50 gloves

x 5 9 units, y 5 6 units

31. Substitution; one variable is already defi ned in terms of the other, so it is easy to substitute one equation into the second; a 5 8, b 5 11.

32. Elimination; because both equations have the same x-term coeffi cient, you can multiply one equation by 21 and add the equations to eliminate the x-term; x 5 94, y 5 21.

33. Elimination; you can multiply the second equation by 22 and add the equations; OR substitution; you can rewrite the second equation in terms of q and substitute into the fi rst equation; p 5 22

3, q 5 203 .

34. Substitution; one variable is already defi ned in terms of the other, so it is easy to substitute one equation into the second; j 5 12, k 5 3.

Name Class Date


15

3-2 Practice Form K


Solve each system by substitution. Check your answers. To start, solve one equation for y and substitute into the other equation.

1. e 4x 1 3y 5 92x 2 y 5 7

2. e 3x 2 y 5 05x 1 2y 5 244

3. e x 2 4y 5 1x 1 2y 5 13

y 5 2x 2 7

4x 1 3(2x 2 7) 5 9

4. Your internet provider off ers two diff erent plans. One plan costs $.02 per email plus a $9 monthly service charge. Th e other plan costs $.05 per email with no service charge.

a. Write a system of equations to model the cost of the two internet plans. b. For how many email messages will both plans cost the same? c. If you send and receive about 500 email messages per month, which plan

should you use?

5. A boat can travel 24 mi in 3 h when traveling with the current. When traveling against the same current, the boat can travel only 16 mi in 4 h. Find the rate of the current and the rate of the boat in still water.

6. Writing Explain how you would solve the system e 2x 1 y 5 10y 5 x 1 4

using substitution. Answers may vary. Sample: Substitute x 1 4 for y in the fi rst equation and solve for x. Then substitute x 5 2 into either of the original equations and solve for y.

x 5 3, y 5 21

x 5 24, y 5 212 x 5 9, y 5 2

Variable may vary. Sample: y 5 0.02x 1 9, y 5 0.05x

300 messages

the plan that costs $.02 per email plus the $9 monthly service charge

2 mi/h, 6 mi/h

Name Class Date


16



7. Error Analysis You and your friend are solving the system e y 5 7x 1 53y 5 21x 1 15

.

You say there are infi nitely many solutions and your friend says the solution is

Q257, 0R. Which of you is correct? What mistake was made?

Solve each system by elimination. Check your answers.

8. e 2x 2 2y 5 422x 1 3y 5 6

9. e 2x 1 4y 5 18x 2 4y 5 6

10. e23x 1 5y 5 163x 1 y 5 8

y 5 10

11. e x 1 3y 5 84x 2 2y 5 4

12. e 3x 2 y 5 26x 1 2y 5 16

13. e 4x 2 5y 5 62x 2 2y 5 4

14. Writing Explain how you would solve the system e 4x 1 2y 5 162x 1 5y 5 10

using elimination.

15. Open-Ended Write a system of equations in which one equation could be multiplied by 22 and the other equation could be multiplied by 3 in order to solve the system using elimination.

Answers may vary. Sample: First, multiply 2x 1 5y 5 10 by 22. Then add the equations to eliminate the variable x. Next, solve for y. Finally, substitute 12 for y into one of the equations and solve for x.

Answers may vary. Sample: e4y 1 3x 5 86y 1 2x 5 6

x 5 12, y 5 10x 5 8, y 5 12 x 5 43, y 5 4

x 5 2, y 5 2 x 5 53, y 5 3 x 5 4, y 5 2

You are correct; your friend forgot to multiply by 3 when substituting the value for y into the second equation.


17

Name Class Date

Multiple Choice

For Exercises 124, choose the correct letter.

Use the system of equations for Exercises 1 and 2. e 14x 2 10y 5 2312x 1 15y 5 12

1. What is the value of x in the solution?

297 2

1528 3

5 34

2. What is the value of y in the solution?

335 3

5 34 24

35

3. Which of the following systems of equations has the solution (4, 21)?

e 3x 2 2y 5 142x 1 2y 5 6

e22x 1 4y 5 623x 1 6y 5 8

e 3x 2 y 5 04x 1 3y 5 26

e 4x 1 9y 5 14x 1 6y 5 22

4. At a bookstore, used hardcover books sell for $8 each and used softcover books sell for $2 each. You purchase 36 used books and spend $144. How many softcover books do you buy?

9 12 18 24

Extended Response

5. A local cell phone company off ers two diff erent calling plans. In the fi rst plan, you pay a monthly fee of $30 and $.35 per minute. In the second plan you pay a monthly fee of $99 and $.05 per minute.

a. Write a system of equations showing the two calling plans. b. When is it better to use the fi rst calling plan? c. When is it better to use the second calling plan? d. How much does it cost when the calling plans are equal?

3-2 Standardized Test PrepSolving Systems Algebraically

D

G

A

I

[4] y 5 0.35x 1 30, y 5 0.05x 1 99 where x 5 number minutes per month and y 5 total plan cost; the fi rst plan is better when you use less than 230 minutes; the second plan is better when you use more than 230 minutes; $110.50 when plans are equal.

[3] correct equations, but with one computational error[2] incorrect equation OR multiple computational errors[1] correct answers, without work shown[0] no answers given


18

Name Class Date

Often, two separate items in a restaurant or a store will be made from essentially the same ingredients. Th e diff erence between two items may involve diff erent proportions of the same ingredients.

Solve each problem by writing two equations using two variables.

1. Each order of lasagna or ravioli at Casa Italia weighs 1 lb and consists of meat fi lling wrapped in pasta. On Tuesday evening, the restaurant served 40 orders of ravioli and 60 orders of lasagna and used 60 lb of meat fi lling. On Wednesday evening, it sold 60 orders of ravioli and 30 orders of lasagna and used 50 lb of meat fi lling.

a. How much meat fi lling is used in an order of ravioli? b. How much meat fi lling is used in an order of lasagna?

2. Foot Friends uses the same type of leather in their men’s and women’s hiking boots. During one week the company used 290 ft2 of leather to make 40 pairs of size 10 men’s hiking boots and 60 pairs of size 7 women’s hiking boots. Another week the company used 275 ft2 of leather to make 50 pairs of size 10 men’s hiking boots and 40 pairs of size 7 women’s hiking boots.

a. How much leather is used to make 1 pair of size 10 men’s hiking boots? b. How much leather is used to make 1 pair of size 7 women’s hiking boots?

3. Th e Fruit Emporium sells a dish of two fl avors of yogurt with one serving of fruit toppings for $1.40. It also sells a dish of three fl avors of yogurt with one serving of fruit toppings for $1.95.

a. How much is one fl avor of yogurt with one serving of fruit toppings? b. How much does a customer pay for the one serving of fruit toppings?

4. A health-food store sells trail mix made with granola and dried fruit. Th e store buys granola at $1.00/lb and dried fruit at $2.00/lb, and it sells these items at a 25% markup. If the trail mix sells for $1.75/lb, what is the recipe for trail mix?

5. Th e same health-food store sells two special mixtures of granola and dried fruit. One customer buys 6 lb of Mixture A and 4 lb of Mixture B and pays $17.00. Another customer reverses the proportions and pays a dollar more.

a. What is the cost per lb of Mixture A? b. What is the recipe for Mixture A? c. What is the cost per lb of Mixture B? d. What is the recipe for Mixture B?

3-2 EnrichmentSolving Systems Algebraically

12 lb

23 lb

3.5 ft2

2.5 ft2

$.85$.30

$1.50

$2.00

35 lb granola and 25 lb dried fruit make 1 lb of trail mix.

45 lb granola and 15 lb dried fruit make 1 lb of Mixture A.

25 lb granola and 35 lb dried fruit make 1 lb of Mixture B.


19

Name Class Date

Follow these steps when solving by substitution.

Step 1 Solve one equation for one of the variables.

Step 2 Substitute the expression for this fi rst variable into the other equation. Solve for the second variable.

Step 3 Substitute the second variable’s value into either equation. Solve for the fi rst variable.

Step 4 Check the solution in the other original equation.

Problem

What is the solution of the system of equations? e 4x 1 3y 5 104x 1 2y 5 10

Step 1 x 5 22y 1 10 Solve one equation for x.

Step 2 4(22y 1 10) 1 3y 5 10 Substitute the expression for x into the other equation. 28y 1 40 1 3y 5 10 Distribute. 25y 5 230 Combine like terms. y 5 6 Solve for y.

Step 3 x 1 2(6) 5 10 Substitute the y value into either equation. x 1 12 5 10 Simplify. x 5 22 Solve for x.

Step 4 4(22) 1 3(6) 0 10 Check the solution in the other equation. 28 1 18 0 10 Simplify. 10 5 10 ✓

Th e solution is (22, 6).

Exercises

Solve each system by substitution.

1. e2x 2 3y 5 22x 1 2y 5 5

2. ea 1 3b 5 4a 5 22

3. e22m 1 n 5 627m 1 6n 5 1

4. e 7x 2 3y 5 21x 1 2y 5 12

3-2 ReteachingSolving Systems Algebraically

x 5 219, y 5 27 a 5 22, b 5 2 m 5 27, n 5 28 x 5 2, y 5 5


20

Name Class Date

Follow these steps when solving by elimination.

Step 1 Arrange the equations with like terms in columns. Circle the like terms for which you want to obtain coeffi cients that are opposites.

Step 2 Multiply each term of one or both equations by an appropriate number.

Step 3 Add the equations.

Step 4 Solve for the remaining variable.

Step 5 Substitute the value obtained in step 4 into either of the original equations, and solve for the other variable.

Step 6 Check the solution in the other original equation.

Problem

What is the solution of the system of equations? e 2x 1 5y 5 2113x 2 2y 5 212

Step 1 2x 1 5y 5 211 Circle the terms that you want to make opposite. 3x 2 2y 5 212

Step 2 6x 1 15y 5 33 Multiply each term of the fi rst equation by 3. 26x 1 14y 5 24 Multiply each term of the second equation by 22.

Step 3 19y 5 57 Add the equations.Step 4 y 5 3 Solve for the remaining variable.

Step 5 3x 2 2(3) 5 212 Substitute 3 for y to solve for x. x 5 22

Step 6 2(22) 1 5(3) 0 11 Check using the other equation. 24 1 15 0 11 11 5 11 ✓

Th e solution is (22, 3). You can also check the solution by using a graphing calculator.

Exercises


5. e 3x 1 2y 5 2173x 2 3y 5 219

6. e25f 1 4m 5 2622f 2 3m 5 21

7. e23x 2 2y 5 526x 1 4y 5 7

8. e22x 2 24y 5 21210x 1 20y 5 210

9. Reasoning Why does a system with no solution represent parallel lines?



If there is no solution, then there are no values of the variables that will make both equations true. This means there is no point that lies on both lines, so the lines never meet and are therefore parallel.

x 5 23, y 5 24 f 5 2, m 5 21 no solution y 5 212x 2 12, where

x is any real number


21

Name Class Date

Th ere are two sets of note cards that show how to solve the system of inequalities below. Th e set on the left explains the thinking. Th e set on the right shows the steps. Write the thinking and the steps in the correct order.

y . ux 2 2 uy # 4

Think Cards Write Cards

Think Write

2 4 6�2�2

2

4

6

xO

y 2 4 6

xO

y

�2�2

2

4

6

2 4 6�2�2

2

4

6

xO

y

3-3 ELL Support Systems of Inequalities

y �

Step 1

Step 2

Step 3

Step 4

First,

Second,

Third,

Fourth,

(2, 2)

2 . u222 u 2 # 42 . 0 2 # 4

✓ ✓

Clearly shade the overlapping region.

Graph the second inequality with a solid line. Shade.

Graph the fi rst inequality with a dotted line. Shade.

Thi k

Pick a point in the overlapping region and check it in both inequalities.

(2, 2)

2 . u222 u 2 # 42 . 0 2 # 4

✓ ✓

2 4 6�2

2

4

6

xO

y

2 4 6�2

2

46

xO

y

graph the fi rst inequality with

graph the second inequality

clearly shade the overlapping

pick a point in the

a dotted line. Shade.

with a solid line. Shade.

region.

overlapping region and check it in both inequalities.

2 4 6�2

2

4

6

xO

y


22

Name Class Date

3-3 Think About a PlanSystems of Inequalities

College Admissions An entrance exam has two sections, a verbal section and a mathematics section. You can score a maximum of 1600 points. For admission, the school of your choice requires a math score of at least 600. Write a system of inequalities to model scores that meet the school’s requirements. Th en solve the system by graphing.

Know

1. Th e sum of the verbal score and the mathematics score must be

2. Each of the scores must be

3.

Need

4. To solve the problem, you need to fi nd

.

Plan

5. What system of inequalities models this situation?

6. Graph your system of inequalities on the grid at the right.

7. How do you know which region in your graph represents the solution?

y

400

0

800

1200(200, 700)

1600

0 400 800 1200 1600

x

less than or equal to 1600.

The mathematics score must be greater than or equal to 600.

Answers may vary. Sample: Let x 5 the verbal score, let y 5 the math score; x L 0,

y L 0, x 1 y K 1600, y L 600.

the possible scores that meet the school’s

requirements

Answers may vary. Sample: Test a point, such as

(200, 700). It satisfi es all of the inequalities, so the

points in its region represent solutions of the problem.

L 0


23

Name Class Date

Find all whole number solutions of each system using a table.

1. e2x 1 2y 5 12x 1 2y # 20

2. e 2x 2 3y $ 12x 1 3y # 21

3. e y , 22x 1 4y # x 1 2

4. e 2x 2 y # 22x 1 y # 5

5. e y . 4x 1 2y 2 4x # 3

6. e y , 2 x3 1 3

2x 1 y $ 4

7. Th e dry cleaner charges $4 to clean a pair of pants and $3 to clean a shirt. You want to get at least 8 items cleaned. You have $32 to spend on dry cleaning.

a. Write a system of inequalities to model the situation. b. Solve the system by using a table.

Solve each system of inequalities by graphing.

8. e y . x 1 2y # 2x 1 1

9. e y # x 1 3y $ x 1 2

10. e x 1 y , 5y , 3x 2 2

11. e 2x 2 2y , 32x 1 2y . 8

12. e23x 1 y , 323x 1 y . 21

13. e 2x 1 2y . 42x 2 2y . 6

14. e 2x $ y 1 32x , 3 2 2y

15. e 3 , 2x 2 yx 2 3y # 4

16. e 2x 1 y . 22x 2 y $ 3

3-3 Practice Form G

Systems of Inequalities

(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)

(0, 4), (0, 5), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1)

4x 1 3y K 32, x 1 y L 8 where x 5 number of pairs of pants and y 5 number of shirts

(0, 8), (0, 9), (0, 10), (1, 7), (1, 8), (1, 9), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (4, 4), (4, 5), (5, 3), (5, 4), (6, 2), (7, 1), (8, 0)

2. (1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (4, 2), (4, 3), (5, 0), (5, 1), (5, 2), (5, 3), (6, 0), (6, 1), (6, 2), (6, 3), (7, 0), (7, 1), (7, 2), (8, 0), (8, 1), (9, 0), (9, 1), (10, 0)

(0, 3)

(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)

(1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1),

(4, 0), (4, 1), (5, 0), (5, 1), (6, 0), (7, 0), (8, 0)

x

y

O

4 64 26

46

46

2

x

y

O

42 6�2�6

46

�4�6

�2x

y

O42�2�4�6

4

�2

8

x

y

O

4 62�2�4�6

42

�4

86

O

2

�2

�2

2

x

y

x

y

O 4�2�4

4

�4�2 8

�6

6

x

y

O 42�2�4

4

�2 8

�6

6

x

y

O

4 62�2�4

42

�2 8

�6

6

xO 2 4�4 �2

�2

�4

4

2

y

(0, 0), (0, 1), (0, 2), (0, 3),


24

Name Class Date

17. Suppose you are buying two kinds of notebooks for school. A spiral notebook costs $2, and a three-ring notebook costs $5. You must have at least 6 notebooks. Th e cost of the notebooks can be no more than $20.



a. Write a system of inequalities to model the situation. b. Graph and solve the system.

18. A camp counselor needs no more than 30 campers to sign up for two mountain hikes.Th e counselor needs at least 10 campers on the low trail and at least 5 campers on the high trail.

a. Write a system of inequalities to model the situation. b. Graph and solve the system.


19. e y , x 2 3y $ u x 2 4 u

20. e22x 1 y . 1y . u x u

21. e y , 23y , 2u x u

22. e y $ 22y # 2u x 1 3 u

23. e y , x 1 3y . u x 2 1 u

24. e y . xy , u x 1 2 u

25. Error Analysis Your homework assignment is to solve the system

e y $ 2y $ u x u

using a graph. You turn in the graph at the right. What

did you do wrong? Draw a correct graph.

26. Open-Ended Write a system of inequalities that has no solution.

27. A doctor needs at least 60 adults for a medical study. He cannot use more than 40 men in the study. Write a system of inequalities to model the situation and solve the system by graphing.

xO�4 �2

�2

�4

2

2

4

4 y

dx 1 y L 62x 1 5y K 20x L 0y L 0

O

4

�44

x

y

O

20

40

20 40 x

yc

x 1 y K 30x L 10y L 5

O

2

4 6

x

y

O

2

�2�2

2

x

y�2 2

�2

�4

xy

�6

O�2

�2�4x

y2

�6 O

2

�2�2

2

x

y

O�2

�2�4 x

y2

xO�4 �2

�2

�4

2

2

4

4 y

You graphed y L u x u as y L x .

Answers may vary, but should be any two inequalities where there is no overlap. Sample: y S x 1 2, y R x

x 1 y L 60, 0 K x K 40, y L 0, where x represents the number of men and y represents the number of women 200

020

Num

ber o

f wom

en

40

60

80

40Number of men

60 80

y

x

Name Class Date


25

3-3 Practice Form K


Find all whole number solutions of each system using a table. To start, make a table of values for x and y that satisfy the fi rst inequality.

1. e y 1 x # 8y 2 1 . 2x

2. e y 2 2x # 22y # 2x 1 5

3. e2y 1 x $ 243y , 29x 1 3

Solve each system of inequalities by graphing. To start, graph the fi rst inequality.

4. e y # 2x 1 2y , 2x 1 1

5. e2x 2 y # 2y 2 2x . 1

6. e y $ x 2 32y , x 1 6

7. You want to bake at least 6 loaves of bread for a bake sale. You want at least twice as many loaves of banana bread as nut bread. Write and graph a system of inequalities to model the situation. Variables may vary. Sample: y L 2x 1 6, y L 2x, where x 5 the number of loaves of nut bread and y 5 the number of loaves of banana bread

x

0 8, 7, 6, 5, 4, 3, 2, 1, 01 7, 6, 5, 4, 3, 2, 1, 02 6, 5, 4, 3, 2, 1, 03 5, 4, 3, 2, 1, 04 4, 3, 2, 1, 05 3, 2, 1, 06 2, 1, 0

y x

01 02 2, 1, 03 4, 3, 2, 1, 04 6, 5, 4, 3, 2, 1, 05 8, 7, 6, 5, 4, 3, 2, 1, 06 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

y x

1 5, 4, 3, 2, 1, 00 4, 3, 2, 1, 0

2 6, 5, 4, 3, 2, 1, 03 7, 6, 5, 4, 3, 2, 1, 04 8, 7, 6, 5, 4, 3, 2, 1, 05 9, 8, 7, 6, 5, 4, 3, 2, 1, 06 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

y

y

xO�2

�2

2

�4

�4 4

y

x

�2�2

2

�4

�4 4O

2

y

Ox

468

1012

4 6 8 10 12

y

O x

2468

4 6 82

Name Class Date


26

8. Writing Explain how you would test whether (25, 9) is a solution of the

system e y . 6x 1 2y # 23x 1 1

.

9. An exam has two sections: a multiple choice section and an essay. You can score a maximum of 100 points. To pass the test, you must get at least 65 points on the essay. Write a system of inequalities to model passing scores. Th en graph the system.

10. For your rock collection display, you want to have at most 25 samples. You want to have at least 3 times as many sedimentary samples as metamorphic samples. Write and graph a system of inequalities to model the situation.


11. cx 2 y # 5x $ 0y $ 0

12. c5x 2 2y . 6x $ 0y $ 0

13. cy , 3x 1 4x $ 1y $ 0



Substitute the x- and y-coordinates into BOTH

inequalities. If BOTH inequalities are true, then the ordered pair is a solution of the system. Since 9 S 228 and 9 K 16 are both true, (25, 9) is a solution of the given system.

y

O x

20406080

40 60 8020

Variables may vary. Sample: y K 2x 1 100, y L 65,where x 5 the score on the multiple choice section and y 5 the score on the essay

y

O x

6121824

4 6 82

Variables may vary. Sample: y L 3x, y K 2x 1 25, where x 5 the number of metamorphic samples and y 5 the number of sedimentary samples

y

Ox

2468

4 6 82

y

Ox

2

4

4 62

y

O x

2468

10

6 8 102 4


27

Name Class Date

Multiple Choice

For Exercises 1−4, choose the correct letter.

1. Which system of inequalities is shown in the graph?

e y # 22x 1 2y . x 2 4

e y $ 22x 1 2y , x 2 4

e y . 22x 1 2y # x 2 4

e y , 22x 1 2y $ x 2 4

2. Which of the following graphs shows the solution of the system of

inequalities? e y $ 22x 1 2y # u3x u

3. Which point lies in the solution set for the system? e y , 5x 2 1y $ 7 2 3x

(25, 1) (2, 23) (4, 4) (1, 6)

4. How many of the ordered pairs in the data table provided are solutions of the

system? e x 1 y # 4x $ 1

6 9 10 15

Short Response

5. Is (4, 22) a solution of the system? e x 1 y . 22x 2 y , 1

Explain how you made your determination.

xO 2 4�4 �2

�2

2

4

�4

y

xO 2 4�4 �2

�2

2

4

�4

y

xO 2 4�4 �2

�2

2

4

�4

y

xO 2 4�4 �2

�2

2

4

�4

y

xO 2 4�4 �2

�2

2

4

�4

y

3-3 Standardized Test PrepSystems of Inequalities

x

0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0

3 1, 0

4 0

y

[2] No; substitute the ordered pair into each inequality and simplify. Both inequalities are false, so the ordered pair is not a solution of the system.

[1] correct answer, without work shown[0] incorrect answer and no work shown OR no answer given

A

G

G

C


28

Name Class Date

Th e regions below were formed by graphing several inequalities. Write the coordinates of each vertex of the fi gure. Study each graph, and determine the inequalities used.

1. Vertices:

Inequalities:

2. Vertices:

Inequalities:

a. For each graph, choose one point in the region. Explain how that point satisfi es each of the inequalities.

b. For each graph, choose a point outside the region that satisfi es one or more of the inequalities but not all of them.

O

4

2

�2

�2

�4�6 4 62

x

y

O

4

6

2

�2

�2

�4

�6

�4�6�8 4 6 82

x

y

3-3 EnrichmentSystems of Inequalities

1a. Answers may vary. Sample: (0, 0); Substituting the values x 5 0 and y 5 0 in each inequality gives a true statement.

2a. Answers may vary. Sample: (0, 0); Substituting the values x 5 0 and y 5 0 in each inequality gives a true statement.

1b. Answers may vary. Sample: (4, 4); 7(4) 1 5(4) 5 48 S 32. However, substituting x 5 4 and y 5 4 in each of the three remaining inequalities gives a true statement.

2b. Answers may vary. Sample: (8, 0); 7(8) 2 8(0) 5 56 S 48. However, substituting x 5 8 and y 5 0 in each of the fi ve remaining inequalities gives a true statement.

(26,22), (23, 5), (1, 5), (6,22);

(28, 1), (28, 6), (0,26), (0, 7), (8, 1), (8, 6);

y L 22, y K 5, 7x 2 3y L 236, 7x + 5y K 32

x L 28, x K 8, 7x 1 8y L 248, 7x 2 8y K 48, x 2 8y L 256, x 1 8y K 56


29

Name Class Date

Solving a System by Using a Table

Problem

An English class has 4 computers for at most 18 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have 4 students and each performance group must have 5 students. In how many ways can you set up the computer groups?

Step 1 Relate the unknowns and defi ne them with variables.

x 5 number of research groups, y 5 number of performance groups number of research groups 1 number of performance groups # 4

4 ? number of research groups 1 5 ? number of performance groups # 18

Step 2 Make a table of values for x and y that satisfy the fi rst inequality. Th e replacement values for x and y must be whole numbers.

Step 3 In the table, check each pair of values to see which satisfy the other inequality. Highlight these pairs. Th ese are the solutions of the system.

You can have:0 groups doing research and 0, 1, 2, or 3 groups watching performances or1 group doing research and 0, 1, or 2 groups watching performances or 2 groups doing research and 0, 1, or 2 groups watching performances or3 groups doing research and 0 or 1 group watching performances or4 groups doing research and 0 groups watching performances

Exercises

Find the whole number solutions of each system using tables.

1. e x 1 y , 4x 1 2y # 10

2. e 6x 2 3y $ 16x 1 3y # 21

3. e x 1 y $ 5y , 22x 1 8

4. e y , 34x 1 2y , 12

x 1 y # 44x 1 5y # 18

3-3 ReteachingSystems of Inequalities

x

0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0

3 1, 0

4 0

y

x

0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0

3 1, 0

4 0

y

(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (3, 0)

(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)

(0, 5), (0, 6), (0, 7), (1, 4), (1, 5), (2, 3)

(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)


30

Name Class Date

Solving a System by Graphing

Problem

What is the solution of the system of inequalities? e 2x 2 y . 12x 1 y $ 3

Step 1 Solve each inequality for y. 2x 2 y . 1 x 1 y $ 32y . 22x 1 1 and y $ 2x 1 3

y , 2x 2 1

Step 2 Graph the boundary lines. Use a solid line for $ or #

xO 2 4�4 �2

�2

2

4

�4

y

inequalities. Use a dotted line for . and , inequalities.

Step 3 Shade on the appropriate side of each boundary line.

xO 2 4�4 �2

�2

2

4

�4

y

Th e overlap is the solution to the system.

Exercises


5. e y # xy $ 3x 2 1

6. e 2x 1 y . 32x 2 y , 2

7. e x . 1y , x 1 1

8. e 2x 1 3y # 92x 2 3y . 1

9. e y , 213x 2 1

y $ 3x 1 1 10. e 4x 1 2y # 1

4x 1 2y # 21

11. e y # 3y . 4x 2 3

12. e 2x 1 y , 33x 2 y , 2

13. e y $ 223x 1 y # 2

3-3 Reteaching (continued) Systems of Inequalities

O

2

�2�2

2x

y

O

2

�2�2

2

x

y

O

2

�2�2

2

x

y

O

2

�2�2

2x

y

O

2

�2�2

2 x

y

O2

�2�2

2

x

y

O

2

�2�2

2x

y

O

2

�2�2

2

x

y

O

2

�2�2

2

x

y


31

Name Class Date

3-4 ELL Support Linear Programming

Choose the word from the list that best matches each sentence.

constraints feasible region linear programming objective function vertices

1. Th e limits or restrictions are also called .

2. is a method for fi nding a minimum or maximum value of some quantity, given a set of constraints.

3. On a graph, the contains all of the points that satisfy the constraints.

4. Th e models a quantity that is related to the constraint variables.

5. Th e of the feasible region are the points where the least and greatest values for the objective function occur.

Use a word from the list above to complete each sentence.

6. Th e inequalities x 2 2y # 4 and 2x 1 y # 6 are the .

7. Th e equation P 5 5x 1 10y is an example of an where P stands for profi t.

8. Th e intersections of the constraint graphs are the of the feasible region.

9. You have to graph the inequalities to fi nd the for a linear programming problem.

Multiple Choice

10. Which of the following is a vertex of the feasible region on the graph at the right?

(5, 4) (2, 7) (3, 6) (1, 1)

11. How many constraints were graphed to form the feasible region on the graph?

2 3 4 5

tex of the feasible region on

Vertex is the singular form of vertices. Vertex means one of the vertices.

constraints

Linear programming

feasible region

objective function

vertices

constraints

objective function

vertices

feasible region

C

H


32

Name Class Date

3-4 Think About a Plan Linear Programming

Cooking Baking a tray of corn muffi ns takes 4 cups of milk and 3 cups of wheat fl our. Baking a tray of bran muffi ns takes 2 cups of milk and 3 cups of wheat fl our. A baker has 16 cups of milk and 15 cups of wheat fl our. He makes $3 profi t per tray of corn muffi ns and $2 profi t per tray of bran muffi ns. How many trays of each type of muffi n should the baker make to maximize his profi t?

Understanding the Problem

1. Organize the information in a table.

2. What are the constraints and the objective function?

Planning the Solution

3. Graph the constraints on the grid at the right.

4. Label the vertices of the feasible region on your graph.

Getting an Answer

5. What is the value of the objective function at each vertex?

6. At which vertex is the objective function maximized?

7. How can you interpret the solution in the context of the problem?

Constraints: Objective Function:

Milk (cups)

Flour (cups)

Profit

Corn MuffinTrays, x

Bran MuffinTrays, y Total

P(0, 5) 5 10; P(3, 2) 5 13; P(4, 0) 5 12; P(0, 0) 5 0

P 5 3x 1 2y

The baker should make 3 trays of corn muffi ns and 2 trays of bran muffi ns to

maximize his profi t.

(3, 2)

e4x 1 2y K 163x 1 3y K 15x L 0y L 0

4x

3x

3x

2y

3y

2y

16

15

3x 1 2y

(0, 5)

(4, 0)(0, 0)(3, 2)

20 4Corn Muffin Trays

6 8

2

0

4

6

8

Bran

Muf

fin T

rays

x

y


33

Name Class Date

3-4 Practice Form G

Linear Programming

Find the values of x and y that maximize or minimize the objective function for each graph.

1. Maximum for 2. Minimum for 3. Maximum for P 5 6x 1 2y P 5 4x 1 y P 5 x 1 y

4. Maximum for 5. Minimum for 6. Minimum for P 5 2x 1 y P 5 x 1 9y P 5 5x 1 10y

Graph each system of constraints. Name all vertices. Th en fi nd the values of x and y that maximize or minimize the objective function.

7. cx 1 2y # 6x $ 2y $ 1

8. cx 1 y # 5x 1 2y # 8x $ 0, y $ 0

9. cx 1 y # 62x 1 y # 10x $ 0, y $ 0

Minimum for Maximum for Maximum for C 5 3x 1 4y P 5 x 1 3y P 5 4x 1 y

x

1A(2,1) B(3,0)

C(0,5)

3 42 5

21

345

y

Ox

1OA(0,0)

B(6,0)

C(2,4)D(0,4)

32 4 5

21

3

y

xB(6,3)

C(0,5)

42 6

4

6

2

y

A(2,3)

Ox

1A(0,0)

B(5,0)

C(4,1)

D(0,3)

32 4

21

345

y

O xB(6,0)

C(0,6)

4 62

4

6

2

y

A( , 2 )23

23

O

x

2A(0,0) B(4,0)

D(0,10)

C(3,4)

6 84 10

42

68

10y

O

(6, 0)

(5, 0)

; (2, 1)

(6, 0)

; (0, 4)

(2, 3)

; (5, 0)

(0, 5) (0, 10)

O1 3 4 5 6

2

4567

x

y

(2, 2)(2, 1)

(4, 1)O1 2 3 4 5

123

567

x

y

(2, 3)

(0, 4)

(5, 0)

(0, 0)O1 2 3 4 6 7

123456

x

y

(5, 0)

(0, 6)

(4, 2)

(0, 0)


34

Name Class Date

10. You are going to make and sell baked goods. A loaf of Irish soda bread is made

with 2 c flour and 14 c sugar. Kugelhopf cake is made with 4 c flour and 1 c

sugar. You will make a profit of $1.50 on each loaf of Irish soda bread and a profit of $4 on each Kugelhopf cake. You have 16 c flour and 3 c sugar.

a. How many of each kind of baked goods should you make to maximize the profit? b. What is the maximum profit?

11. Suppose you make and sell skin lotion. A quart of regular skin lotion contains 2 c oil and 1 c cocoa butter. A quart of extra-rich skin lotion contains 1 c oil and 2 c cocoa butter. You will make a profit of $10/qt on regular lotion and a profit of $8/qt on extra-rich lotion. You have 24 c oil and 18 c cocoa butter.

a. How many quarts of each type of lotion should you make to maximize your profit?

b. What is the maximum profit?

Graph each system of constraints. Name all vertices. Th en fi nd the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

12. c3x 1 2y # 62x 1 3y # 6x $ 0, y $ 0

13. c4x 1 2y # 42x 1 4y # 4x $ 0, y $ 0

14. cx 1 y # 54x 1 y # 8x $ 0, y $ 0

Maximum for Maximum for Minimum for P 5 4x 1 y P 5 3x 1 y C 5 x 1 3y


Linear Programming

15. Writing Explain why solving a system of linear equations is a necessary skill for linear programming.

16. A doctor allots 15 minutes for routine offi ce visits and 45 minutes for full physicals. Th e doctor cannot do more than 10 physicals per day. Th e doctor has 9 available hours for appointments each day. A routine offi ce visit costs $60 and a full physical costs $100. How many routine offi ce visits and full physicals should the doctor schedule to maximize her income for the day? What is the maximum income?36 routine offi ce visits; 0 full physicals; $2160

(2, 0); 8 (1, 0); 3 (0, 0); 0

Four loaves of Irish soda bread and two Kugelhopf cakes

2.5 qt regular, 1 qt extra-rich$33

$14

See graphs below.

O1 2 3 5 6 7

1

3

567

x

y

(2, 0)

(0, 2)65

65

,

O

1 2 3

2

3

x

y

(1, 0)

(0, 1)23

23

,

O4 6 71 3 5

21

4

6

3

5

7

x

y

(0, 5)

(1, 4)

(2, 0)

12. 13. 14.

15. Answers may vary. Sample: You must know how to fi nd the point of intersection to be able to determine the critical points of a system of constraints. The vertices enable you to fi nd the x and y values that maximize or minimize an objective function.

See below.

Name Class Date


35

3-4 Practice Form K

Linear Programming


1. cy # 2 x 1 3y # 2

12

x 1 2x $ 0, y $ 0

2. cy # 2x 1 4y # 2

13x 1 2

x $ 0, y $ 0 3. cy #

12

x 1 2y # 2x 1 8x $ 2, y $ 1

Maximum for Minimum for Maximum for P 5 24x 1 3y P 5 2x 1 3y P 5 x 2 4y

4. Teams chosen from 30 forest rangers and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week.

a. Write an objective function and constraints for a linear program that models the problem.

b. How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted in a week?

Number of Teams

Number of Rangers

Number of Trainees

Number of Trees Planted

ExperiencedTeams

TrainingTeams Total

x

2x

0

500x

y

y

2y

200y

x 1 y

30

16

500x 1 200y

y

Ox

B(0, 2) C(2, 1)

D(3, 0)A(0, 0)

�2

�4

4

2�2�4

y

Ox

�2

4

6

42�2

(0, 2) (3, 1)

(4, 0)(0, 0)

y

O x

6

2

4 62A(2, 1)

B(2, 3)C(4, 4)

D(7, 1)

max P at (0, 2) min P at (0, 0) max P at (7, 1)

P 5 500x 1 200y; constraints: 2x 1 y K 30, 2y K 16, x L 0, y L 0

15 experienced teams and 0 training teams; 7500 trees planted

Name Class Date


36

Graph each system of constraints. Name all vertices. Th en fi nd the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

5. cy # 23x 1 72y 1 x # 9x $ 0, y $ 0

6. cy 2 5 # 4x

y 1 x # 10x $ 0, y $ 3

7. c3y # 2x 1 9y 1 2x # 8x $ 0, y $ 0

Minimum for Maximum for Maximum for P 5 2x 1 y P 5 7x 2 5y P 5 4x 1 y

8. Reasoning Why are x $ 0 and y $ 0 part of the constraints in many linear programs?

9. Error Analysis Your friend says the graph to the right can be used to determine the maximum for P 5 2x 1 3y with the

constraints c2y 1 x # 10y 2 12 # 24xx $ 0, y $ 0

. What mistake did your friend make?


Linear Programming

y

O x

4

8

12

8 124

y

x2

4

6

4 62

(0, 4½) (1, 4)

(0, 0) O

y

O x

2

4

4 62

A(0, 3) D(7, 3)

B(0, 5)

C(1, 9) y

(0, 0)

(0, 3)

x

2

4

6

4 62(4, 0)

(3, 2)

O

min P at (0, 0) 5 0 max P at (7, 3) 5 34 max P at (4, 0) 5 16

In most real-life situations, you cannot have negative resources.

He shaded the graph to show the constraints as greater than instead of less than.

See graphs below.

5. 6. 7.


37

Name Class Date

Multiple Choice


1. Th e vertices of a feasible region are (0, 0), (0, 2), (5, 2), and (4, 0). For which objective function is the maximum cost C found at the vertex (4, 0)?

C 5 22x 1 3y C 5 2x 1 7y C 5 4x 2 3y C 5 5x 1 3y

2. A feasible region has vertices at (0, 0), (3, 0), Q32, 72R , and (0, 3). What are the

maximum and minimum values for the objective function P 5 6x 1 8y?

minimum (0, 0) 5 0 minimum (0, 0) 5 14

maximum Q32, 72R 5 37 maximum Q3

2, 72R 5 17

minimum (0, 0) 5 0 minimum (0, 0) 5 0

maximum (3, 0) 5 24 maximum (0, 3) 5 30

3. Which values of x and y minimize N for the objective function N 5 2x 1 y?

Constraints cx 1 y $ 8x 1 2y $ 14x $ 0, y $ 0

(0, 0) (0, 7) (2, 6) (8, 0)

4. Which of the following systems has the vertices (0, 5), (1, 4), (3, 0), and (0, 0)?

cx 1 y $ 52x 1 y $ 6x $ 0, y $ 0

cx 1 y # 52x 1 y # 6x $ 0, y $ 0

cx 1 y # 5x 1 2y # 6x $ 0, y $ 0

cx 1 y # 52x 1 2y # 6x $ 0, y $ 0

Short Response

5. Th e fi gure at the right shows the feasible region for a system of constraints. Th is system includes x $ 0 and y $ 0. What are the remaining constraints? Show your work.

x

B(10,20)

C(0,40)

A(50,0)10

10

20

30

40

20 30 40 50

y

O

3-4 Standardized Test Prep Linear Programming

C

F

A

G

[2] Work should show the use of two points of each line

to fi nd slope and linear inequalities, b2x 1 y L 40x 1 2y L 50

[1] correct inequalities, without work shown OR correct process with one computational error

[0] no answer given and no work shown OR no answer given


38

Name Class Date

Th e Supreme Shipping Company can load its trucks with both rectangular and cylindrical containers. A rectangular container has a volume of 100 ft3 and weighs 200 lb. A cylindrical container has a volume of 200 ft3 and weighs 100 lb. Let x denote the number of rectangular containers carried by a truck, and let y denote the number of cylindrical containers.

1. What constraint must be satisfied if each truck has room for at most 4200 cubic ft of containers?

2. What constraint must be satisfied if each truck can carry a maximum of 4800 lb?

3. What additional constraints must be satisfied because the problem involves real objects?

4. Graph the feasibility set on the grid and label its vertices. Call the vertex on the x-axis A, the vertex on the y-axis B, and the vertex in Quadrant I C. Label the origin O.

5. Suppose that Supreme Shipping charges $60 to ship either a rectangular or a cylindrical container and wishes to maximize its income.

a. What is the objective function? b. What is the value of the objective function at vertex A? c. At vertex B? d. At vertex C? e. What combination of containers should Supreme Shipping use to maximize

its income?

3-4 EnrichmentLinear Programming

100x 1 200y K 4200

200x 1 100y K 4800

I 5 60x 1 60y$1440

$1260$1800

x L 0, y L 0, where x and y are whole numbers

18 rectangular containers,12 cylindrical containers

O

12

24

36

48

36 4812 24

x

y

C

A

B


39

Name Class Date

Problem

What point in the feasible region maximizes P for the objective function P 5 10x 1 15y? What point minimizes P?

Constraints dx 1 y # 163x 1 6y # 60x $ 0y $ 0

Step 1 Step 2 Step 3

Graph the constraints and shade the feasible region.

Find the coordinates for each vertex of the region.

Evaluate P at each vertex.

VERTEX P 5 10x 1 15y

A (0, 0) P 5 10(0) 1 15(0) 5 0

B (16, 0) P 5 10(16) 1 15(0) 5 160

C (12, 4) P 5 10(12) 1 15(4) 5 180

D (0, 10) P 5 10(0) 1 15(10) 5 150

Th e maximum value of the objective function is 180. It occurs when x 5 12 and y 5 4.

Th e minimum value of the objective function is 0. It occurs when x 5 0 and y 5 0.

Exercises


1. d5y 1 4x # 355y 1 x $ 20y # 6x $ 1

2. dx 1 y $ 2x $ y

x # 4y $ 0

3. c3x 1 4y $ 125x 1 6y # 301 # x # 3

P 5 8x 1 2y P 5 x 1 3y P 5 x 2 2y

3-4 ReteachingLinear Programming

xO

y

D

C

BA

8

8

16

vertices: (1, 6), Q1, 195 R ,

Q54, 6R , (5, 3)

max: 46 at (5, 3);

min: 785 at Q1, 19

5 R

vertices: (4, 0), (1, 1), (2, 0), (4, 4)

max: 16 at (4, 4); min: 2 at (2, 0)

vertices: Q1, 94R , Q1, 25

6 R , Q3, 3

4R , Q3, 52R

y

x2

O

4

6

4 62

(5, 3)

(1, 6)(5/4, 6)

(1, 19/5)

y

x2

O

4

6

4 62(2, 0)

(4, 4)

(4, 0)

(1, 1) y

x2

O

4

6

4 62

(1, 9/4)

(1, 3/6)

(3, 5/2)

max: 32 at Q3, 34R ;

min: 2 223 at Q1, 25

6 R


40

Name Class Date

Problem

Your school band is selling calendars as a fundraiser. Wall calendars cost $48 per case of 24. You sell them at $7 per calendar. Pocket calendars cost $30 per case of 40. You sell them at $3 per calendar. You make a profi t of $120 per case of wall calendars and $90 per case of pocket calendars. If the band can buy no more than 1000 total calendars and spend no more than $1200, how can you maximize your profi t if you sell every calendar? What is the maximum profi t?

Relate Organize the information in a table.

Defi ne Let x 5 number of cases of wall calendars Let y 5 number of cases of pocket calendarsWrite Use the information in the table and the defi nitions of x and y to write the constraints

and the objective function. Simplify the inequalities if necessary. 24x 1 40y # 1000 48x 1 30y # 1200 23x 1 45y # 125 8x 1 5y # 200

c3x 1 5y # 1258x 1 5y # 200x $ 0, y $ 0

Objective function: P 5 120x 1 90y

Step 1 Step 2 Step 3Graph the constraints and shade to see the feasible region.

Find the coordinates for each vertex of the region.

Evaluate the objective function using the vertex coordinates.

A(0, 0) P 5 120(0) 1 90(0) 5 0

B(25, 0) P 5 120(25) 1 90(0) 5 3000

C(15, 16) P 5 120(15) 1 90(16) 5 3240

D(0, 25) P 5 120(0) 1 90(25) 5 2250

You can maximize your profi t by selling 15 cases of wall calendars and 16 cases of pocket calendars. Th e maximum profi t is $3240.

Exercises 4. Your band decides to sell the wall calendars for $9 each. a. How many of each type of calendar should you now buy to maximize your

profi t? b. What is the maximum profi t?

4

48

12162024

8 12 16 20 24

y

xO


Linear Programming

Number of Cases

Number of Units

Cost

Profit

Wall Calendars

x

24x

48x

120x

Pocket Calendars

y

40y

30y

90y

Total

1000

1200

120x 1 90y

25 cases of wall calendars and no cases of pocket calendars$4200


41

Name Class Date

3-5 ELL Support Systems With Three Variables

A student solved the system of equations •x 2 3y 1 3z 5 242x 1 3y 2 z 5 154x 2 3y 2 z 5 19

and wrote a description of

each step on note cards. Write each description next to the correct step.

x 2 3y 1 3z 5 24

2x 1 3y 2 z 5 15

3x 1 2z 5 11

2x 1 3y 2 z 5 15

4x 2 3y 2 z 5 19

6x 2 2z 5 34

3x 1 2z 5 11

6x 2 2z 5 34

9x 5 45

x 5 5

3(5) 1 2z 5 11

2z 5 24

z 5 22

5 2 3y 1 3(22) 5 24

23y 5 23

y 5 1

2(5) 1 3(1) 2 (22) 5 15

15 5 15 ✓

4(5) 2 3(1) 2 (22) 5 19

19 5 19 ✓

Th e solution is (5, 1, 22).

Write the solution as an ordered triple.

Write the two new equations as a system. Solve for x.

Substitute the value for x into one of the equations and solve for z.

Pair the equations. Eliminate y in each system.

Check your solution by substituting values for x, y, and z into the other two original equations.

Substitute the values for x and z into one of the original equations and solve for y.

Write the solution as an ordered triple.

Pair the equations. Eliminate y in each system.

Write the two new equations as a system. Solve for x.

Substitute the value for x into one of the equations and solve for z.

Substitute the values for x and z into one of the original equations and

solve for y.

Check your solution by substituting values for x, y, and z into the other two

original equations.


42

Name Class Date

3-5 Think About a PlanSystems With Three Variables

Sports A stadium has 49,000 seats. Seats sell for $25 in Section A, $20 in Section B, and $15 in Section C. Th e number of seats in Section A equals the total number of seats in Sections B and C. Suppose the stadium takes in $1,052,000 from each sold-out event. How many seats does each section hold?

Understanding the Problem

1. Defi ne a variable for each unknown in this problem.

Let x 5

Let y 5

Let z 5

2. What system of equations represents this situation?

Planning the Solution

3. Can you write a simpler equivalent equation for one of the equations in your system? If so, write the equivalent equation.

4. What method of solving looks easier for this problem? Explain.

Getting an Answer

5. Solve the system of equations.

6. How can you interpret the solution in the context of the problem?

the number of seats in Section B

the number of seats in Section A

x 1 y 1 z 5 49,000

x 5 y 1 z

25x 1 20y 1 15z 5 1,052,000

the number of seats in Section C

yes; 5x 1 4y 1 3z 5 210,400

(24,500, 14,400, 10,100)

Answers may vary. Sample: Substitution; one of the equations is already

Section A holds 24,500 seats, Section B holds 14,400 seats, and Section C

holds 10,100 seats.

solved for x.


43

Name Class Date

3-5 Practice Form G

Systems With Three Variables

Solve each system by elimination. Check your answers.

1. •x 1 y 1 z 5 212x 2 y 1 2z 5 252x 1 2y 2 z 5 4

2. •2x 1 2y 1 2z 5 32x 2 2y 1 2z 5 63x 1 2y 2 2z 5 13

3. •2x 1 y 5 9x 2 2z 5 232y 1 3z 5 15

4. •x 2 y 1 2z 5 102x 1 y 2 2z 5 53x 2 3y 1 6z 5 22

5. •2x 2 y 1 z 5 243x 1 y 2 2z 5 03x 2 y 5 24

6. •2x 2 y 2 z 5 42x 1 2y 1 z 5 13x 1 y 1 z 5 16

7. •x 1 5y 1 5z 5 210x 1 5y 1 5z 5 2x 1 2y 1 3z 5 23

8. •x 2 y 2 z 5 0x 2 2y 2 2z 5 322x 1 2y 2 z 5 3

9. •3x 1 2y 1 2z 5 63x 2 2y 1 2z 5 143x 1 3y 2 3z 5 26

10. •2x 1 2y 1 3z 5 222x 1 2y 2 3z 5 113x 2 2y 1 3z 5 4

11. •2x 2 5y 1 z 5 32x 1 2y 2 2z 5 2122x 1 2z 5 6

12. •2x 1 3z 5 23x 1 6y 5 63x 2 2z 5 8


13. •3x 1 y 2 z 5 03x 2 y 1 z 5 45x 1 z 5 7

14. •2x 2 2y 5 12x 1 3y 1 z 5 02x 2 2z 5 18

15. •x 1 y 1 4z 5 522x 1 2z 5 33x 1 y 2 2z 5 0

16. •23x 1 2y 1 2z 5 426x 1 4y 2 2z 5 2929x 2 2y 1 2z 5 10

17. •2x 2 3y 1 z 5 23x 2 5y 1 7z 5 211210x 1 4y 2 6z 5 28

18. •2x 1 3y 1 2z 5 282x 2 3y 2 2z 5 62x 2 3y 1 2z 5 21

19. •14x 2 3y 1 5z 5 21513x 1 2y 2 6z 5 1017x 2 2y 1 4z 5 25

20. •5x 2 3y 1 2z 5 394x 1 4y 2 3z 5 343x 2 2y 1 6z 5 14

21. •x 1 y 1 z 5 62x 2 y 1 2z 5 62x 1 y 1 3z 5 10

22. •2x 1 y 2 z 5 33x 2 y 1 3z 5 32x 2 3y 1 2z 5 3

23. •22x 2 3y 1 3z 5 422x 1 3y 2 3z 5 2426x 2 9y 1 3z 5 12

24. •2x 1 y 2 z 5 12x 1 2z 5 32x 1 2y 5 4

No unique solution

(0, 5, 0)

(4, 0, 21)

(8, 21, 22)

(3, 3, 3)

(1, 2, 3)

No solution (21, 1, 21) (4, 1, 3)

(5, 21, 22) (2, 22, 2) (23, 22, 21)

(2, 21, 23) (4, 21, 22)

(1, 1, 2) (3, 1, 26) Q21, 4, 12R

Q23, 212, 11

2R (22, 21, 22) (21, 23, 24)

(22, 0, 5)

(2, 23, 22) No unique solution (1, 1, 1)


44

Name Class Date

Write and solve a system of equations for each problem.

25. Th e sum of three numbers is 22. Th e sum of three times the fi rst number, twice the second number, and the third number is 9. Th e diff erence between the second number and half the third number is 10. Find the numbers.

26. Monica has $1, $5, and $10 bills in her wallet that are worth $96. If she had one more $1 bill, she would have just as many $1 bills as $5 and $10 bills combined. She has 23 bills total. How many of each denomination does she have?

27. Writing How do you decide whether substitution is the best method to solve a system in three variables?

28. Error Analysis A student solves the system of equations. •2x 1 3y 2 3z 5 132x 1 3y 1 3z 5 475x 2 3y 1 3z 5 1

Th e student gets a solution of (2, 12, 3). Is the solution correct? How can you be sure? Show your work.

29. Reasoning Why is there no solution to the system? •22x 2 3y 1 2z 5 522x 2 3y 1 2z 5 2224x 1 6y 2 2z 5 10

30. Th e fi rst number plus the third number is equal to the second number. Th e sum of the fi rst number and the second number is six more than the third number. Th ree times the fi rst number minus two times the second number is equal to the third number. What is the sum of the three numbers?

31. Which of the following is a system with the solution (6, 22, 23)?

•12x 1 4y 2 2z 5 515x 1 4y 1 2z 5 1615x 1 6y 2 2z 5 12

•22x 2 2y 1 2z 5 1623x 2 3y 1 2z 5 924x 1 3y 2 2z 5 220

•24x 2 3y 1 2z 5 3123x 2 2y 1 2z 5 823x 1 2y 2 4z 5 213

•22x 1 2y 2 2z 5 2723x 1 5y 1 2z 5 222x 1 3y 1 4z 5 230



Answers may vary. Sample: If one equation can be solved easily for one variable, then substitution is the best method to use.

Yes; check the solution by substituting it in the original equations; show the values substituted into the equations.

Answers may vary. Sample: All three equations have the same relative coeffi cients and variables, but end

8

D

•3x 1 2y 1 z 5 223x 1 2y 1 z 5 93x 1 2y 2 1

2z 5 10,

•x 1 5y 1 10z 5 96x 1 1 5 y 1 zx 1 y 1 z 5 23

,

where x represents the fi rst number, y represents the second number, and z represents the third number; 3, 5, 210

where x represents the number of $1 bills, y represents the number of $5 bills, and z represents the number of $10 bills; eleven $1 bills, seven $5 bills, fi ve $10 bills

up equaling different values. This is a model for parallel planes. 2x 23y 1 z cannot equal 5, 22, and 25 at the same time. Therefore, there is no solution to this system.

Name Class Date


45

3-5 Practice Form K


Solve each system by elimination. Check your answers. To start, pair the equations to eliminate one variable and add.

1. cx 1 2y 1 z 5 102x 2 y 1 3z 5 252x 2 3y 2 5z 5 27

2. c2a 1 b 1 c 5 9a 1 2b 1 c 5 8a 1 b 1 2c 5 11

22(x 1 2y 1 z) 5 (22 ? 10) 1 2x 2 y 1 3z 5 25

25y 1 4z 5 225


3. c2x 1 y 1 z 5 142x 2 3y 1 2z 5 224x 2 6y 1 3z 5 25

4. c3x 1 2y 2 z 5 1224x 1 y 2 2z 5 4x 2 3y 1 z 5 24

y 5 22x 2 z 1 14 2x 2 3(22x 2 z 1 14) 1 2z 5 22

2x 1 6x 1 3z 2 42 1 2z 5 22 5x 1 5z 5 40

x 1 z 5 8

5. You have 17 coins in pennies, nickels, and dimes in your pocket. Th e value of the coins is $0.47. Th ere are four times the number of pennies as nickels. How many of each type of coin do you have?

6. Writing When you solve a system of equations, explain how you can determine if your solution is correct.

(7, 4, 25)

(1, 5, 7)

(2, 1, 4)

(2, 0, 26)

12 pennies, 3 nickels, 2 dimes

Substitute your solution back into the original equations. If all of the equations are true, the solution is correct.

Name Class Date


46

Solve each system using any method.

7. c4x 2 y 1 z 5 022x 1 2y 1 3z 5 32x 1 3y 2 2z 5 219

8. cx 1 2y 1 z 5 22022x 1 y 2 z 5 255x 1 2y 2 z 5 16

9. c3x 1 2y 1 2z 5 132x 1 y 2 z 5 25x 2 3y 1 z 5 216

10. For a party, you are cooking a large amount of stew that has meat, potatoes, and carrots. Th e meat costs $6 per pound, the potatoes cost $3 per pound, and the carrots cost $1 per pound. You spend $48.50 on 13.5 pounds of food. You buy twice as many carrots as potatoes.

a. Write a system of three equations that represent how much food you bought.

b. How much of each ingredient did you buy?

11. Multiple Choice What is the value of z in the solution of the system?

c3x 2 4y 1 2z 5 20x 1 y 2 z 5 246x 2 y 1 2z 5 23

2 5

21 25



(22, 25, 3)

(4, 27, 210)

C

(23, 6, 5)

Variables may vary.

Sample: cx 1 y 1 z 5 13.56x 1 3y 1 z 5 48.5z 5 2y6 lb meat, 2.5 lb potatoes, 5 lb carrots


47

Name Class Date

Gridded Response

Solve each exercise and enter your answer in the grid provided.

1. A change machine contains nickels, dimes, and quarters. Th ere are 75 coins in the machine, and the value of the coins is $7.25. Th ere are 5 times as many nickels as dimes. How many quarters are in the machine?

2. Th e sum of three numbers is 23. Th e fi rst number is equal to twice the second number minus 7. Th e third number is equal to one more than the sum of the fi rst and second numbers. What is the fi rst number?

3. A fi sh’s tail weighs 9 lb. Its head weighs as much as its tail plus half its body. Its body weighs as much as its head and tail. How many pounds does the fi sh weigh?

4. You are training for a triathlon. In your training routine each week, you bike 5 times as far as you run and you run 4 times as far as you swim. One week you trained a total of 200 miles. How many miles did you swim that week?

5. Th ree multiplied by the fi rst number is equal to the second number plus 4. Th e second number is equal to one plus two multiplied by the third number. Th e third number is one less than the fi rst number. What is the sum of all three numbers?

Answers

1. 2. 3. 4. 5.

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

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9876543210

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9876543210

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9876543210

–

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

–

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

–

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

–

3-5 Standardized Test PrepSystems With Three Variables

1 5

1

5

5

5

7 2

7

2

8

8

1 0

10


48

Name Class Date

3-5 EnrichmentSystems With Three Variables

The Italian Navigator Has LandedTh e above phrase was one of the most important coded messages that has ever been sent. It referred to the fact that a team of physicists had managed to achieve the fi rst successful controlled nuclear chain reaction. Th e physicist who directed these eff orts was an accomplished theorist and experimenter, whose work in producing artifi cial radioactive elements won him the Nobel Prize in Physics in 1938.

1 2 3 4 5 6 7 8 9 10 11 12

First solve each of the following sets of equations. For each letter with a value between 1 and 12, write that letter in its corresponding location in the puzzle.

•A 1 C 2 2E 5 2A 2 2C 5 5A 2 C 1 E 5 4

Solution:

•23L 1 M 1 N 5 423L 2 M 1 5N 5 2423L 1 4N 5 17

Solution:

•3A 1 B 1 2O 5 13A 2 B 1 2O 5 93A 2 B 2 2O 5 25

Solution:

•3A 2 2I 1 F 5 43A 2 2I 1 F 5 03A 1 2I 1 F 5 12

Solution:

•2B 1 I 1 K 5 82I 2 K 5 24B 1 6I 5 20

Solution:

•22D 1 R 5 024D 1 2E 1 R 5 724D 1 3E 2 2R 5 3

Solution:

•4A 1 2U 1 I 5 25A 2 3U 1 2I 5 175A 2 5U 5 3

Solution:

•3L 1 4R 1 2T 5 223L 2 4R 2 2T 5 75L 1 4R 1 3T 5 23

Solution:

E N R I C O F E R M I

A 5 15, C 5 5, E 5 9

L 5 23, M 5 11, N 5 2

A 5 22, B 5 23, O 5 6

A 5 0, I 5 4, F 5 8

No unique solution

D 5 25, E 5 1, R 5 10

A 5 22, U 5 21, I 5 12

L 5 0, R 5 3, T 5 25


49

Name Class Date

Problem

What is the solution of the system?

•x 1 2y 1 3z 5 216

2x 2 2y 1 3z 5 2192x 1 2y 1 2z 5 219

Use elimination. Th e equations are numbered to make the process easy to follow.

x 1 2y 1 2z 5 2162x 1 2y 1 2z 5 219

3y 1 3z 5 215

2x 2 2y 1 3z 5 19

2x 1 2y 1 2z 5 29

2x 2 4y 1 3z 5 21922x 1 4y 1 4z 5 218

3y 1 7z 5 227

3y 1 (23z) 5 21523y 1 (27z) 5 227

24z) 5 212z) 5 213

3y 1 3(3) 5 2153y 5 216

y 5 212x 1 2 1 3 5 216

x 5 211

Th e solution is (1, 2, 3).

ExercisesSolve each system by elimination. Check your answers.

1. •2x 2 3y 1 2z 5 104x 1 2y 2 5z 5 104x 2 3y 1 5z 5 8

2. •3x 2 3y 1 2z 5 62x 1 3y 1 2z 5 23x 1 5y 1 4z 5 4

3. •6x 2 4y 1 5z 5 315x 1 2y 1 2z 5 135x 1 4y 1 5z 5 2

4. •3x 1 2y 1 3z 5 24x 2 2y 1 3z 5 242x 1 2y 1 2z 5 8

5. •5x 1 2y 1 3z 5 53x 2 3y 2 3z 5 93x 1 2y 1 4z 5 6

6. •4x 1 3y 1 2z 5 214x 1 3y 1 2z 5 2102x 2 4y 2 2z 5 26

A

B

C

A

C

Pair the equations to eliminate x .

B

C

D

B

C

E

Multiply equation 3 by 2 to eliminate x . Then add the two equations.

D

E

Equations 4 and 5 form a system.

Multiply equation 5 by 21 and add to equation 4 to eliminate y and solve for z.

A

D Substitute z 5 3 into equation 4 and solve for y .

Substitute the values of y and z into one of the original equations. Solve for x .

3-5 ReteachingSystems With Three Variables

Pair a different set of equations.

(23, 22, 4)

(4, 2, 2) (2, 22, 2) (3, 22, 1)

(21, 3, 2) (2, 24, 3)


50

Name Class Date

Problem

What is the solution of the system? Use substitution. •24x 1 3y 2 2z 5 1924x 2 2y 1 3z 5 823x 1 2y 1 2z 5 15

Step 1 Choose an equation that can be solved easily for one variable. Choose equation 1 and solve for x.

Step 2 Substitute the expression for x into equations 2 and 3 and simplify.

4(23y 1 2z 1 19) 2 2y 1 3z 5 8 23(23y 1 2z 1 19) 1 2y 1 2z 5 15212y 1 8z 1 76 2 2y 1 3z 5 8 9y 2 6z 2 57 1 2y 1 2z 5 15

D 214y 1 11z 5 268 E 11y 2 4z 5 72

Step 3 Write the two new equations as a system. Solve for y and z. e214y 1 11z 5 268211y 2 14z 5 272

256y 1 44z 5 2272 Multiply D by 4. 214y 1 11z 5 268 Substitute y 5 8 into D.121y 2 44z 5 2792 Multiply E by 11. 214(8) 1 11z 5 268

65y 5 520 11z 5 44y 5 8 z 5 4

Step 4 Use one of the original equations to solve for x.x 1 3y 2 2z 5 19 Substitute y 5 8 and z 5 4 into A.

x 1 3(8) 2 2(4) 5 19x 5 3

Th e solution of the system is (3, 8, 4).

Exercises


7. •24x 1 3y 2 2z 5 72x 2 2y 1 3z 5 152x 1 2y 2 2z 5 26

8. •22x 2 2y 1 5z 5 2523x 1 2y 2 7z 5 1022x 2 3y 1 6z 5 212

9. •2x 2 4y 1 3z 5 153x 2 4y 2 2z 5 72x 1 2y 1 5z 5 22

10. •22x 1 2y 2 2z 5 2122x 1 2y 1 2z 5 1124x 2 3y 1 2z 5 4

11. •2x 2 3y 1 2z 5 72x 2 5y 2 4z 5 212x 1 4y 2 6z 5 22

12. •23x 2 4y 2 3z 5 21322x 1 4y 2 5z 5 2525x 1 2y 1 5z 5 3

A

B

C

A x 1 3y 2 2z 5 19

x 5 23y 1 2z 1 19

D

E



(22, 7, 11)

(1, 2, 3)

(3, 4, 1)

(6, 1, 2)

(5, 1, 2)

(2, 4, 5)


51

Name Class Date

3-6 ELL Support Solving Systems Using Matrices

Choose the word from the list that best matches each sentence.

coeffi cient column dimensions matrix matrix element row

1. A rectangular array of numbers is called a .

2. Th e numbers that are written horizontally in a matrix form a .

3. Each number in a matrix is called a .

4. Th e of a matrix are the number of rows and columns in the array.

5. A in a matrix is formed by the numbers written vertically.

6. You can use a to represent a system of equations.

7. Each and constant from a system of equations are used to represent the system within a matrix.

8. Each matrix represents an equation.

9. Th e last matrix shows the constants to the right of the equal signs.

Th e number of rows is always the fi rst number.

Th e number of columns is always the second number.

Th e size of a matrix is described by listing the dimensions of the matrix. Th is means listing the number of rows and columns in the matrix.

3 rows cC25 20 21

29 22S Th e size of this matrix is 3 3 2.

2 columnss

Write the dimensions of each matrix below.

10. c21 5 0 263 23 8 2

d

11. C5 22 9 90 6 23 18 8 7 24

S 12. c22 9 0 26 d

2 3 4

3 3 4

1 3 4

matrix

row

matrix element

dimensions

column

matrix

coeffi cient

row

column


52

Name Class Date

3-6 Think About a PlanSolving Systems Using Matrices

Paint A hardware store mixes paints in a ratio of two parts red to six parts yellow to make two gallons of pumpkin orange. A ratio of fi ve parts red to three parts yellow makes two gallons of pepper red. A gallon of pumpkin orange sells for $25, and a gallon of pepper red sells for $28. Find the cost of 1 quart of red paint and the cost of 1 quart of yellow paint.

Know

1. Th ere are z z quarts in 1 gallon.

2. z zqt red 1z zqt yellow 5z zqt pumpkin orange

z zqt red 1z zqt yellow 5z zqt pepper red

3. z zquarts of pumpkin orange cost z z.z zquarts of pepper red cost z z.

Need

4. To solve the problem you need to defi ne:

5. To solve the problem you need to fi nd:

Plan

6. What system of equations represents this situation?

7. How can you represent the system of equations with a matrix?

8. Solve the system of equations using the matrix.

9. How can you interpret the solutions in the context of the problem?

4

2

4

4

5

6

3

8

8

c1 35 3

` 2556d

$25

$28

Variables may vary. Sample: x 5 cost of 1 qt of red paint, y 5 cost of 1 qt of yellow paint

Variables may vary. Sample: x 1 3y 5 25, 5x 1 3y 5 56

(7.75, 5.75)

1 qt of red paint costs $7.75 and 1 qt of yellow paint costs $5.75.

a system of two equations


53

Name Class Date

Identify the indicated element.

A 5 c3 54 1

` 86d B 5 £

0 6 34 5 12 2 0

† 2 13210

§

1. A13 2. B24 3. B12 4. A22

5. B31 6. A21 7. B23 8. A11

Write a matrix to represent each system.

9. e23x 1 4y 5 2422x 1 4y 5 7

10. e26x 5 1123x 1 4y 5 2

11. c4x 2 2y 1 2z 5 105x 1 2y 2 3z 5 04x 2 3y 1 3z 5 6

Write the system of equations represented by each matrix.

12. D 2 5 023 1 2

4 0 23 4 13

65T 13. c2 1

0 4 ` 27

9d 14. D6 4 22

1 25 20 3 21

4 17210

0T

Solve the system of equations using a matrix.

15. e24x 2 5y 5 1022x 1 5y 5 4

16. c x 2 2y 1 3z 5 18 9x 1 2y 2 z 5 2226x 2 y 1 2z 5 4

17. c 3x 2 4y 1 z 5 1522x 2 6y 1 3z 5 4 2x 1 2y 2 2z 5 21

3-6 Practice Form G

Solving Systems Using Matrices

c23 122 4

` 2427d

c22x 1 5y 5 1323x 1 y 1 2z 5 624x 2 3z 5 5

(3, 2)(1, 24, 3) (4, 12, 5)

c 6 023 4

` 112d

e2x 1 y 5 274y 5 9

D4 21 25 2 231 23 1

4 1006T

c6x 1 4y 2 2z 5 176x 2 5y 1 2z 5 210 3y 2 z 5 0

8

2

13

4

6

1

1

3


54

Name Class Date

Graphing Calculator Solve each system.

18. c4x 1 y 2 2z 5 32y 1 z 5 43x 2 3y 2 z 5 9

19. c5x 2 2y 1 z 5 212x 2 y 2 2z 5 53x 1 2y 1 2z 5 2

20. c3x 1 5z 5 2422x 1 y 2 3z 5 92x 2 2y 1 9z 5 0

21. Suppose the movie theater you work at sells popcorn in three diff erent sizes. A small costs $2, a medium costs $5, and a large costs $10. On your shift, you sold 250 total containers of popcorn and brought in $1726. You sold twice as many large containers as small ones.

a. How many of each popcorn size did you sell? b. How much money did you bring in from selling small size containers?

22. Open Ended Write a matrix for a system of equations that does not have a unique solution.

23. Th e following matrix shows the prices passengers on an airline fl ight paid for a recent ticket and how many passengers were on that fl ight. Some passengers paid full price for their tickets, and some bought their tickets during a half-price sale. How many passengers bought each price of ticket?

c 1 1120 240

` 10020,160

d

24. Error Analysis Your friend says that the matrix below represents the system of equations. What error did your friend make? What is the correct system of equations?

D 4 0 2123 2 22

1 23 22 4 4

2226T c4x 1 y 2 z 5 4

23x 1 2y 2 2z 5 22x 2 3y 2 2z 5 26



(4, 21, 6) (2, 3, 25) (23, 6, 1)

32 people bought $120 tickets; 68 people bought $240 tickets.

68 small, 46 medium, 136 large$136

Answers may vary. Sample: c 4 2222 1

` 263d .

Your friend incorrectly converted the 0 matrix entry to a coeffi cient of 1 when writing the

fi rst equation; c4x 2 z 5 423x 1 2y 2 2z 5 22x 2 3y 2 2z 5 26

Name Class Date


55

3-6 Practice Form K


Identify the indicated element.

A 5 C4 22 21 4 10 5 27

S 1. a21 2. a13 3. a32 4. a33 row 2, column 1


5. b2x 1 y 5 8x 2 3y 5 23

6. b24x 1 y 5 232x 1 y 5 0

7. c3x 1 2y 2 2z 5 95x 1 y 2 3z 5 27x 1 4y 1 3z 5 5

Write the system of equations represented by each matrix.

8. B21 24 3

` 521R 9. B1 6

2 4 ` 7

22R 10. C21 5 24

3 4 123 6 27

† 0

212S

1 2 5 27

B2 11 23

` 823R

b2x 1 2y 5 54x 1 3y 5 21

B24 12 1

` 230R

bx 1 6y 5 72x 1 4y 5 22

C3 2 225 1 231 4 3

† 9

275S

c2x 1 5y 2 4z 5 03x 1 4y 1 z 5 2123x 1 6y 2 7z 5 2

Name Class Date


56

11. Error Analysis Your classmate says that in the matrix to the right, a23 is 6. What mistake did your classmate make? What is the correct answer?

Solve the system of equations using a matrix.

12. bx 2 2y 5 21022x 2 3y 5 21

13. b23x 2 y 5 214x 1 y 5 3

14. b2x 1 5y 5 2112x 1 y 5 2

B 1 2222 23

` 21021R

15. You work at a fruit stand that sells apples for $2 per pound, oranges for $5 per pound, and bananas for $3 per pound. Yesterday you sold 60 pounds of fruit and made $180. You sold 10 more pounds of apples than bananas.

a. Write a matrix to show the system of equations for this situation. b. How many pounds of each kind of fruit did you sell yesterday? c. What kind of fruit did you sell the most?

16. Open-Ended Write a matrix for a system of two equations that does not have a solution.

17. Writing Explain how to write the system b5x 1 2y 5 34x 5 7

as a matrix.



A 5

3 21 4C1 5 222 6 23

SShe used column 2, row 3 instead of row 2, column 3. The correct answer is 22.

First, write each equation in the same variable order. Line up the variables. Leave space where a coeffi cient is 0. Write the matrix using the coeffi cients and constants. Enter zeros

for any “missing” variables. The matrix is B5 24 0

` 37R .

Answers may vary. Sample: any matrix containing the equations of two parallel lines

28 lb apples, 14 lb oranges, 18 lb bananas

(24, 3)

(2, 25) (23, 21)

C1 1 12 5 31 0 21

† 60

18010S

apples


57

Name Class Date

Multiple Choice


1. Which system of equations is equivalent to D4 21 23 0 41 5 3

4 627T?

c4x 1 y 1 2z 5 63x 1 4z 5 2x 1 5y 1 3z 5 7

c4x 1 y 1 2z 5 63x 1 y 1 4z 5 2x 1 5y 1 3z 5 7

c4x 2 y 1 2z 5 63x 1 4z 5 2x 1 5y 1 3z 5 7

c4x 2 y 1 2z 5 63x 1 y 1 4z 5 2x 1 5y 1 3z 5 7

2. What is the solution of the system represented by the matrix D 2 3 2123 24 2

1 2 21 4 2

223T?

(1, 3, 4) (4, 23, 1) (24, 3, 21) (3, 24, 21)

3. How many elements are in a 2 3 3 matrix? 2 4 5 6

Short Response

4. A clothing store is having a sale. A pair of jeans costs $15 and a shirt costs $8. You spend $131 and buy a total of 12 items. Using a matrix, how many pairs of jeans and shirts do you buy? Show your work.

3-6 Standardized Test PrepSolving Systems Using Matrices

[2] Matrices may vary. Sample: c 1 115 8

` 12131

d ; 5 pairs of jeans; 7 shirts

[1] correct solution, without work shown OR correct process with one computational error

[0] incorrect answer and no work shown OR no answer given

B

H

D


58

Name Class Date

Well-Conditioned Systems of Linear EquationsA system of linear equations is said to be well-conditioned if a small change in the values of the coeffi cients produces a small change in the values of the solutions. A system is said to be ill-conditioned if a small change in the values of the coeffi cients produces a large change in the values of the solutions.

To determine whether a system of linear equations is well-conditioned or ill-conditioned, change each coeffi cient by one percent. Each time, write and solve the new system, fi nding the values of x and y to two decimal places by writing and solving a matrix. Th en compare the new values of x and y to the solutions of the original system. A change of less than one percent in the values of x and y can be considered small.

1. Determine whether System 1 is well-conditioned or ill-conditioned by completing the following steps.

System 1: e 3x 1 2y 5 50003x 2 2y 5 3000

a. Find the values of x and y. b. Change the coeffi cient of x in the fi rst equation by one percent,

from 1 to 1.01. Write the new system. Find the values of x and y. c. Change the coeffi cient of y in the fi rst equation by one percent, from

1 to 0.99. Write the new system. Find the values of x and y. d. Change the coeffi cient of x in the second equation by one percent,

from 3 to 3.03. Write the new system. Find the values of x and y. e. Change the coeffi cient of y in the second equation by one percent,

from 22 to 22.02. Write the new system. Find the values of x and y. f. Is the system well-conditioned or ill-conditioned?

2. Determine whether System 2 is well-conditioned or ill-conditioned by completing the same steps as in Exercise 1.

System 2: e 3x 1 2y 5 250013x 2 2y 5 29997

3-6 Enrichment Solving Systems Using Matrices

1a. 2600, 2400 2a. 1, 5000

1b. e1.01x 1 y 5 50003x 2 2y 5 3000

; 2589.64, 2384.46 2b. e1.01x 1 y 5 50013x 2 2y 5 29997

; 1.00, 4999.99

1c. ex 1 0.99y 5 50003x 2 2y 5 3000

; 2609.66, 2414.49 2c. ex 1 0.99y 5 50013x 2 2y 5 29997

; 21.12, 5030.18

1d. e x 1 y 5 50003.03x 2 2y 5 3000

; 2584.49, 2415.51 2d. e x 1 y 5 50013.03x 2 2y 5 29997

; 0.99, 5000.01

1e. e x 1 y 5 50003x 2 2.02y 5 3000

; 2609.56, 2390.44 2e. e x 1 y 5 50013x 2 2.02y 5 29997

; 20.92, 4980.08

1f. well-conditioned 2f. ill-conditioned


59

Name Class Date

Problem

How can you represent the system of equations with a matrix? c24x 2 3y 1 5z 5 21324x 1 3y 5 322x 1 4y 1 3z 5 17

Step 1 Write each equation in the same variable order. Line up the like variables. Write in variables that have a coeffi cient of 0.

c24x 2 3y 1 5z 5 21324x 1 3y 1 0z 5 21322x 1 4y 1 3z 5 217

Step 2 Write the matrix using the coeffi cients and constants. Remember to enter a 1 for variables with no numeric coeffi cient.

D 4 23 51 3 0

22 4 3 4 213

317T

Exercises


1. e 2x 1 y 5 233y 5 5

2. c3x 2 5y 1 2z 5 94x 1 7y 1 z 5 32x 2 z 5 12

3. c5x 2 y 1 3z 5 23y 1 2z 5 64x 1 3y 1 z 5 1

4. c2x 2 z 5 35y 1 4z 5 252x 1 2y 5 1

5. cz 5 6x 1 y 5 23x 2 2y 2 5z 5 10

6. c2z 2 5x 1 3y 5 42y 1 2x 1 4z 5 223x 1 z 2 2y 5 25

3-6 ReteachingSolving Systems Using Matrices

c2 10 3

` 235d

D 2 0 210 5 4

21 2 0 4 3

251T

D3 25 24 7 12 0 21

4 93

12T

D0 0 11 1 03 22 25

4 62

10T

D5 21 30 3 24 3 1

4 261T

D25 3 22 21 43 22 1

4 42225T


60

Name Class Date

Problem

What is the solution of the system? e 2x 1 5y 5 52x 1 2y 5 27

Step 1 Write the matrix for the system.

c 2 521 2

` 527d

Step 2 Multiply Row 2 by 2. Add to Row 1. Replace Row 1 with the sum. Write the new matrix.

2 5 5)12(21 2 27)

0 9 29) c 0 9

21 2 ` 29

27d

Step 3 Divide Row 1 by 9. Write the new matrix.

19(0 9 29) c 0 1

21 2 ` 21

27d

Step 4 Multiply Row 1 by 22. Add to Row 2. Replace Row 2 with the sum. Write the new matrix.

22(0 1 21)121 2 27)

21 0 25) c 0 1

21 0 ` 21

25d

Step 5 Multiply Row 2 by 21. Write the new matrix.

21(21 0 25) c0 11 0

` 215d

Th is matrix is equivalent to the system e y 5 21x 5 5

. Th e solution is (5, 21).

Exercises

Solve each system of equations using a matrix.

7. e24x 1 3y 5 62x 2 y 5 21

8. e 6x 1 y 5 222x 1 3y 5 13

9. e 3x 1 2y 5 2424x 2 3y 5 7

3-6 Reteaching (continued) Solving Systems Using Matrices

(21, 4) (2, 25)(3, 22)


61

Name Class Date

Do you know HOW?

Solve each system of equations by graphing.

1. e y 5 22x 1 5x 2 y 5 1

2. b2x 1 y 5 103y 5 2x 1 6

Solve each system of equations by substitution or elimination.

3. by 5 x

x 2 4y 5 0 4. b

5x 1 y 5 025x 1 2y 5 30

5. b2x 1 y 2 11 5 03x 2 y 5 21


6. b2x 2 3y , 9x 1 y . 22

7. b2x 2 y $ 1y # 2|x 2 3|21

8. Pump Up Gym has an initial joining fee of $205 and monthly membership dues of $15. Universe Gym has an initial joining fee of $125 and monthly membership dues of $19.

a. When will the costs to join and maintain membership at the gyms be equal?

b. If you planned on continuing your gym membership for only 2 years, which gym would you join? Explain.

Do you unDerstanD?

9. Open-Ended Write and solve a system of two inequalities by graphing.

10. Error Analysis Your teacher asked the class to graph the system of equations to find the solution. You turned in Graph A and said there were no solutions. Your friend turned in Graph B and said there were infinite solutions. Which of you is correct? What mistake was made?

e 3y 5 6x 2 1225y 5 210x 2 20

Chapter 3 Quiz 1 Form G

Lessons 3-1 through 3-3

xO 2 44 2

2

2

4

4

yGraph A

xO 2 44 2

2

2

4

4

yGraph B

x

y

O 2 44 22

4

4

2

yx

O 22

2

4

4 6

(0, 0)

(2, 1) (3, 4)

(22, 10)(2, 7)

at 20 months

Pump Up Gym costs less over 2 years, but it has a much higher initial fee.

Answers may vary. Sample: x 1 y S 0; y K 0

Answers may vary. Sample: You are correct. Your friend forgot to change the negative sign on the second equation’s y-intercept when dividing by 25.

x

y

O

1 321

21

3

12

3 2

3


62

Name Class Date

Chapter 3 Quiz 2 Form G


Do you know HOW?

Graph the system of constraints. Find the values of x and y that maximize or minimize the objective function for each graph.

1. cx 1 y # 82x 1 y # 10x $ 0, y $ 0

2. cy $ 2x 2 1y # 2x 1 2x $ 23

3. cx 2 y # 5x 1 y # 5x $ 21, y # 2

Maximum for Minimum for Maximum for C 5 80x 1 20y P 5 3x 1 4y C 5 x 1 3y

Solve each system by substitution or elimination.

4. c3x 2 3y 1 z 5 2924x 2 3y 2 z 5 02x 2 y 1 2z 5 4

5. c6x 2 y 5 12x 1 3y 1 2z 5 924x 1 2y 1 z 5 2

6. c4x 2 2y 2 3z 5 2124x 1 y 1 2z 5 23x 1 y 1 z 5 27


7. e 6x 1 5y 5 2622x 1 y 5 11

8. c23x 1 4y 2 2z 5 64x 2 5y 2 4z 5 122x 1 y 1 3z 5 212

9. c4x 2 z 5 225x 1 7y 1 z 5 32x 2 5y 1 2z 5 11

Do you unDerstanD?

10. Open Ended Give an example of a system of three equations in three variables that has (2, 21, 4) as a solution. Check your answers to show the ordered triple is a solution for all three equations.

11. A fruit market is selling oranges in a 5 lb bag for $6 and a 10 lb bag for $10. You spend $68 and buy a total of 8 bags of oranges. Using a matrix, how many 5 lb bags and 10 lb bags of oranges did you buy? How many total pounds of oranges did you buy?

check as equal when (2, 21, 4) is substituted for the variables.

three 5 lb bags; five 10 lb bags; 65 lb

C 6 522 1

†2611S C

23 4 224 25 24

22 1 3 †

61

212S C

4 0 2125 7 1

2 25 2 †

23

11S

(5, 0); maximum value is 400

(23, 27); minimum value is 237

(3, 2); maximum value is 9

Any three equations that all

(23, 2, 6) (1, 5, 24) (22, 4, 25)


63

Name Class Date

Do you know HOW?

Without graphing, classify each system as independent, dependent, or inconsistent.

1. e 2x 2 15 5 2y

2y 2 10 5 22x 2. e 2x 5 y 2 7

4x 2 2y 1 4 5 0


3. e y 5 2x 1 8y 5 3x 2 1

4. e 2x 2 y 5 22x 2 2y 5 4

5. e2x 1 y 5 22x 1 y 5 21

Graph the solutions of each system.

6. e y . x 2 53x 1 y # 22

7. e y # x 1 2y . |x 2 3| 1 1

8. You have 10 fewer quarters than dimes and 5 fewer nickels than quarters. The total value of the coins is $4.75. How many quarters, nickels, and dimes do you have?

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.

9. cx # 3y # 7x $ 0, y $ 0

10. c2x 1 y # 30x 1 y # 20x $ 0, y $ 0

Maximum for Minimum for P 5 2x 1 3y C 5 x 1 4y

11. Which point gives the minimum value for P 5 3x 1 2y and lies within the system of restrictions?

c1 # x # 62 # y # 5x 1 y # 10

Chapter 3 Chapter Test Form G

(9, 26) (0, 22) (21, 1)

dependent inconsistent

xy

O

1 321

1

2

3 2

345

O 2 4 6 8

2

4

6

8

x

y

O

2

4

6

62 4

x

y

O

4

2 4 6 8 101214161820

81216202428323640

x

y

10 quarters, 20 dimes, 5 nickels

(1, 2)

(0, 0), (3, 0), (3, 7), (0, 7); (3, 7), 27

(0, 0), (0, 20), (10, 10), (15, 0); (0, 0), 0


64

Name Class Date

12. What is the solution of the system represented by the matrix?

C2 4 231 22 273 5 0

† 23

53S

Solve each system of equations.

13. c5x 1 4y 2 z 5 12x 2 2y 1 z 5 12x 2 y 1 z 5 2

14. cx 1 y 1 z 5 02x 1 3y 1 2z 5 21x 2 y 1 z 5 2

15. cx 1 2y 5 04x 2 z 5 45y 1 z 5 21

Do you unDerstanD?

16. List three methods used to solve systems of equations. Describe the strengths of each method.

17. You burn 4 Cal/min walking and 10 Cal/min running. You walk 10 to 20 min each day and run 30 to 45 min each day. You never spend more than an hour running and walking together. How much time should you spend on each activity to maximize the number of Calories you burn? Will you have exercised enough to burn off a 500 Calorie meal?

18. Plumber A charges $25 for a house call and $50 for each hour spent on the job. Plumber B charges $35 for a house call and $45 for each hour spent on the job. If your job will take 4 hours to complete, which plumber should you use? How much will it cost you?

19. Open Ended Write a system of inequalities that has infinite solutions.

20. Error Analysis A student says that the system of equations is represented by the matrix. What error did the student make? What is the correct matrix?

c5x 2 2y 1 2z 5 73x 1 4y 5 112x 2 6y 1 5z 5 5

C5 22 23 4 12 26 5

† 7

115S

Chapter 3 Chapter Test (continued) Form G

(6, 23, 1)

The student used a 1 for the z-coefficient in the second equation instead of a 0.

Plumber B; $215

Answers may vary. Sample: 2x 2 3y K 7, 3y 2 2x L 27

yes, you will have burned off 510 Calories.45 min running, 15 min walking;

(0, 1, 3) No unique solution

Graphing, substitution, elimination; Answers may vary. Sample: Graphing

(2, 21, 4)

C5 22 23 4 02 26 5

† 7

115S

allows you to visualize the solution. Substitution enables you to reduce the number of variables quickly. Elimination is applicable to any system and efficient with two variables.

Name Class Date


65

Chapter 3 Quiz 1 Form K


Do you know HOW?

Solve each system by graphing.

1. e y 5 3x 1 4y 5 3x 2 1

2. e 6x 1 3y 5 2922x 1 3y 5 15

3. e 4x 1 y 5 4y 2 x 5 4


4. e 2x 2 y 5 28y 5 24x 1 2

5. e 2x 2 y 5 53x 1 y 5 5

6. e 3x 2 2y 5 23x 1 4y 5 50

Do you unDerstanD?

7. The community theater is selling tickets to its play. An adult ticket costs $12 and a child ticket costs $8. The theater wants to take in at least $2720 from ticket sales and has only 275 seats.

a. Write a system of inequalities to model the situation. b. What is one possible combination of ticket sales that would satisfy the

theater’s goal?

8. Reasoning Is it possible for a dependent linear system to consist of two lines with different slopes?

O

y

x

22

4

4

4 2 4 O

y

x

22

4

2

46

O

y

x

2

4

6

2

2 4

no solution

(21, 6)

(23, 3)

(2, 21)

(0, 4)

(6, 8)

Variables may vary. Sample: ex 1 y K 27512x 1 8y L 2720

Answers may vary. Sample: 180 adult tickets and 70 child tickets

No; a dependent system has two lines whose graphs are the same. If the lines have different slopes, then their graphs are not the same.

Name Class Date


66

Chapter 3 Quiz 2 Form K


Do you know HOW?

Find the values of x and y that maximize or minimize the objective function for each graph.

1. •y # 24x 1 92y # 2x 1 11x $ 0, y $ 0

2. •y # 22x 1 52y # 28x 1 16x $ 0, y $ 0

3. •y # 21

3 x 1 7

y # 22x 1 12x $ 0, y $ 0

Maximum for Minimum for Maximum for P 5 2x 2 5y P 5 x 1 3y P 5 22x 1 7y


4. •5x 1 3y 2 z 5 2122x 2 y 1 3z 5 44x 1 2y 1 z 5 21

5. •2x 2 2y 1 4z 5 283x 1 y 2 4z 5 162x 2 3y 1 z 5 7

6. •2x 2 y 1 3z 5 26x 1 y 2 5z 5 1324x 1 3y 2 z 5 4

Do you unDerstanD?

7. Error Analysis To represent the system •2x 2 y 1 4z 5 11x 1 2y 2 6z 5 2113x 2 10z 5 5

, you picked Matrix A.

Your friend picked Matrix B. Which of you is correct? What mistake was made?

Matrix A Matrix B

C2 21 41 2 263 0 210

† 11

2115S C

2 21 41 2 263 1 210

† 11

2115S

8. The sum of three numbers is 21. The second number is two more than twice the first number. The second number is three times the third number. What are the numbers?

(2.25, 0)

(23, 5, 1)

(0, 0)

(4, 0, 21)

(0, 7)

(1, 2, 22)

You are correct; your friend used 1 as the coefficient of a missing variable instead of 0.

first number: 5, second number: 12, third number: 4

Name Class Date


67

Chapter 3 Test Form K

Do you know HOW?


1. e 2x 2 y 5 76x 2 3y 5 14

2. e 5x 1 2y 5 1226x 2 2y 5 214

3. e 5x 1 2y 5 284x 1 3y 5 2

Graph the solution of each system.

4. e y $ 22x 1 3y , x

5. e y 2 1 # 3xy 1 1 # x

6. e y . 2x 1 4y # 2x 1 1

7. You have 13 bills in your wallet in $1, $5, and $10 bills. There are twice as many $1 bills as $5 bills. The number of $10 bills is one more than the number of $5 bills. How many of each bill do you have? How much money do you have?

Graph the system of constraints. Identify all vertices. Find the values of x and y that maximize or minimize the objective function. Then find the maximum or minimum value.

8. •y # 2x 1 74y # x 1 8x $ 0, y $ 0

Maximize for P 5 5x 1 2y

y

x

22

2

4

4 4

y

x

2

2

4

4 42

y

x

2

2

4

6

42

y

x

4

6

4 6A(0, 0)

B(0, 2)

D(7, 0)

C(4, 3)

2

no solution (2, 1) (24, 6)

six $1 bills, three $5 bills, four $10 bills; $61

max P at (7, 0) 5 35

Name Class Date


68

Chapter 3 Test (continued) Form K

9. What is the solution of the system represented by the matrix? C2 21 23 2 21

21 23 2 †

120

11S

(24, 1, 3) (1, 24, 3) (3, 24, 1) (1, 3, 24)

Do you unDerstanD?

10. Writing Explain how you determine whether a system of linear equations is independent, dependent, or inconsistent without graphing the lines.

11. Mechanic A charges $45 for car repairs and $80 for each hour spent on your car. Mechanic B charges $60 for repairs and $60 for each hour spent on your car.

a. If your car takes 5 hours to repair, which mechanic charges the least money?

b. How much will it cost you to have the work done by the less expensive mechanic?

12. At a bookstore, you spend $76 on 11 books and magazines. Books cost $8 each and magazines cost $5 each. Write a matrix that represents this system. How many books and how many magazines did you buy?

13. Reasoning The sum of three numbers is 15. The second number is twice the third number. Do you have enough information to determine the three numbers? If so, what are the three numbers? If not, what information do you still need? No; you need a third equation that defines another relationshipbetween two or three of the numbers.

C

Rewrite both equations in slope-intercept form. If the lines have the same slope and same y-intercept, then they are equations of the same line, and the system is dependent. If the lines have the same slope but different y-intercepts, they are parallel lines, and the system is inconsistent. If the lines have different slopes, then the system is independent.

Mechanic B

$360

B1 18 5

` 1176R ; 7 books, 4 magazines

Name Class Date


69

task 1

Make three systems of two equations/inequalities from any in the box.

y 5 2x 1 1y 5 5x 2 11y $ 22x 1 4

2x 1 2y 5 2y 5 21

3x 1 3y 5 3x 1 1

y # 2x 1 4y 5 2x 2 1x 1 y 5 1

y 5 26x

y $ 0

Use each method to solve one system. Show your work.

a. graphing b. substitution c. elimination

Then find a system that has each of the following.

d. coincident lines e. intersecting lines f. parallel lines g. perpendicular lines

Explain your reasoning. Your models should present situations in which you make comparisons and draw conclusions.

Chapter 3 Performance Tasks

task 2

Your art club wants to sell greeting cards using members’ drawings. Small blank cards cost $10 per box of 25. Large blank cards cost $15 per box of 20. You make a profit of $52.50 per box of small cards and $85 per box of large cards. The club can buy no more than 350 total cards and spend no more than $210. a. How can the art club maximize its profit? b. The card company has a minimum order requirement of 5 boxes of each size per order.

Does the art club meet this minimum requirement when it maximizes its profit? c. What is the maximum profit the art club can make and meet the minimum order

requirement?

[4] Check students’ work.[3] Student writes, solves, and graphs systems and clearly demonstrates an in-depth

understanding of the mathematical principles involved. A comparison is made, and the answer fully supports the conclusion.

[2] Student shows a solid understanding of the mathematical principles. A comparison is made, but further detail or more clarity is needed.

[1] Student shows a limited understanding of the mathematical principles involved. Situation does not present a comparison or draw a conclusion.

[0] Student makes no attempt, or no response is given.

[4] a. 25x 1 20y " 350, 10x 1 15y " 210, x # 0, y # 0, maximize profit for P 5 52.50x 1 85y, 14 boxes of large cards, 0 boxes of small cards

b. No, this does not meet the minimum order requirements c. An order of 6 boxes of small cards and 10 boxes of large cards meets the minimum

requirements and gives a maximum profit of $1165.[3] correct equations, but with one computational error[2] incorrect equations OR multiple computational errors[1] student began writing inequalities but did not solve or show understanding[0] no attempt was made to solve the problem

Name Class Date


70

task 3Explain in detail how to solve a system of equations with three variables by each method.

a. graphing b. substitution c. elimination

Using each method, solve the following system.

cx 5 7 y 2 3z 5 10x 2 8z 5 23

Chapter 3 Performance Tasks (continued)

task 4

You want to put together 20 kits to make beaded necklaces using 3 kinds of beads: silver, black, and crystal. Each necklace requires a total of 30 beads. Silver beads cost $6 for a package of 24 beads. Black beads cost $7.20 for a package of 48 beads. Crystal beads cost $8 for a package of 16 beads. Each necklace has 4 times as many black beads as crystal beads. You have $138 to spend on beads. a. How do the equations at the right relate to the problem? What does each

variable represent?

cx 1 y 1 z 5 30

0.25x 1 0.15y 1 0.50z 5 13820

y 5 4z

b. Write a matrix to represent the system and solve. What does this solution tell you? c. How many packages of each kind of bead should you buy to make 20 necklace

kits? Is $138 enough money to buy this many packages of beads?

[4] Check students’ work; (7, 4, 22)[3] Student’s explanation of each method is accurate and

contains sufficient detail to indicate a clear understanding

[4] a. The first equation shows the number of beads needed for each necklace; the second equation shows the cost per bead for each kind of bead and the amount to spend for each necklace; the third equation shows that there are 4 times as many black beads as crystal beads in each necklace.

b. C1 1 1

0.25 0.15 0.50 1 24

† 306.9

0S ; x 5 10 silver beads, y 5 16 black beads, z 5 4 crystal beads

c. You need 9 packages of silver beads, 7 packages of black beads, 5 packages of crystal beads; no, this many packages costs $144.40.

[3] Showed understanding of equation relations and variables in problem; set up correct matrix, but with one computational error.

[2] incorrect relations OR incorrect matrix OR multiple computational errors[1] Student began explaining relations OR began writing matrix, but did not solve or show

understanding.[0] Student made no effort to solve the problem.

of the method. Student solves system correctly using each method. Student shows steps with enough detail to indicate understanding.

[2] Student’s explanation of each method is accurate and contains sufficient detail to indicate a clear understanding of the method. Student solves system correctly using all three methods. Steps could be explained more clearly or have more detail.

[1] Student’s explanation does not explain in detail each method. Student solves the system correctly using only two methods. Explanation of the steps is not clear or missing important details.

[0] Student makes no attempt, or no response is given.


71

Name Class Date

Multiple ChoiceFor Exercises 1–10, choose the correct letter.

1. Which line is parallel to y 5 32x 1 2

3?

y 532

x 1 6 y 5 3x 123 y 5

23

x 132 y 5

63

x 123

2. Which is a solution of the system of inequalities? e y 2 4 . 0y ≥ x 1 1

(3, 3) (21, 2) (1, 5) (0, 2)

3. What point in the feasible region maximizes C for the objective function C 5 4x 2 2y?

(9, 0) (22, 220) (25, 22) (0, 29)

4. Which number is rational but not an integer?

1,000,000 "16 11 0.25

5. The data in a scatter plot can be modeled by the equation y 5 2x. What kind of correlation is this?

horizontal negative vertical positive

6. What is the value of the expression 3a2 2 2a 1 1 evaluated at a 5 22? 27 9 17 215

7. Which of the following best describes the transformations used to obtain the graph of y 5 2 |x 1 1| 2 3 from the graph of y 5 | x |?

Reflect across the x-axis, shift right one unit and down three units.

Reflect across the x-axis, shift left one unit and down three units.

Reflect across the y-axis, shift left one unit and up three units.

Reflect across the y-axis, shift right one unit and down three units.

8. What is the solution of the inequality 2(3 1 2x) 1 3 $ 22(3 1 x)?

all real numbers no real numbers x $ 2 52 x #25

9. What is the slope of the line through the points (5, 24) and (23, 7)?

118 2

118 2

811 8

11

10. Which graph shows the system represented by the matrix? c 0 324 21

` 95d

xO4 2

24

2 4

24

y

xO4 2

2

4

2

2

4

4 y

O4 2 2

4

8

12

x

y

xO4 2

2

4

2

2

4

4 y

Chapter 3 Cumulative Review

A

H

A

I

B

H

B

H

B

G


72

Name Class Date

Short Response

11. What is the slope of each line?

a. 2x 2 5y 5 3 b. 3x 1 2y 5 4

12. For each function what is f (22)?

a. f (x) 5 3x 1 5 b. f (x) 5 2 52

x 2 2

Chapter 3 Cumulative Review (continued)

13. A store sells pens and notepads in packages. The price of a package of 8 pens and 3 notepads is $4.25. The price of a package of 15 pens and 10 notepads is $11.25. The store is also running a sale on a package of 1 pen and 1 notepad for $1.10. Is the sale price a better deal than the other packages? Show your work.

14. Suppose y varies directly with x, and x 5 215 when y 5 18. What is x when y 5 212? Show your work.

15. Solve the system of equations by substitution. Show your work. e 5x 2 3y 5 214x 2 y 5 2

Extended Response

16. A musician wants to sell CDs of her music and DVDs of her concert videos. It costs $.60 to make a CD and $3 to make a DVD. The profit for each CD is $9.25 and the profit for each DVD is $12. The musician can spend no more than $1170 and she wants to have at least 750 items to sell.

a. How many CDs and DVDs should the musician sell to maximize her profit? b. The musician wants to buy a new amplifier system that costs $7000. Will

she have enough money to make this purchase after selling everything? c. If the musician is able to make this purchase, how much money will she have left

over? If she is not able to make this purchase, how much more money does she need?

[4] a. 0.60x 1 3y " 1170, x 1 y # 750, x # 0, y # 0, maximize profit for P 5 9.25x 1 12y, 1950 CDs, 0 DVDs to maximize profit

b. Yes, she can make the purchase c. She will have $11,037.50 left over.[3] correct equations, but with one computational error[2] incorrect equations OR multiple computational errors[1] correct answers, without work shown[0] no answers given

No; in the original packages a pen is $.25 and a notepad is $.75. Together they are $1. In the sale package, they are $1.10.

25

21 3

2 32

(1, 2)

y 5 kx; 18 5 k(215); k 5 18215 5 26

5;

y 5 2 65 x; 212 5 2

65 x; x 5 212Q2

56R 5 10

y 5 4x 2 2; 5x 2 3(4x 2 2) 5 21; 5x 2 12x 1 6 5 21; 27x 5 27; x 5 1; y 5 4x 2 2 5 4(1) 2 2 5 2; (1, 2)


73

T E A C H E R I N S T R U C T I O N S

About the ProjectThe Chapter Project gives students an opportunity to simulate a successful business by minimizing costs, maximizing profits, and establishing a process for filling orders promptly. Students research costs, study profit margins, and establish selling prices. Students track inventory by designing a spreadsheet.

Introducing the Project• Encouragestudentstokeepallproject-relatedmaterialsinaseparatefolder.

• Askstudentstographasystemofinequalities.Investigatehowsolutionstothis system could be used to maximize profits.

• Discusstheinformationneededtokeeptrackofacompany’sfilledordersand available stock, and the best way to organize that information on a spreadsheet.

activity 1: Graphing Students write and graph a system of inequalities. Then they determine how many pints of each sauce they can make.

activity 2: analyzingStudents use their answers from Activity 1 to determine how much of each sauce they should produce to maximize profit. Students use this information to find the maximum profit.

activity 3: researchingStudents visit a grocery store to estimate the cost of each ingredient of the sauces.Students use this information to find the cost to produce one pint of each type of sauce. Then students determine the selling price that will maintain the same profit margin.

activity 4: OrganizingStudents determine a production schedule to meet demand given that the company will produce the same amount of sauce each week. Students prepare a spreadsheet to keep track of filled orders and available stock.

Finishing the ProjectYou may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their results.

• Havestudentsreviewtheirprojectworkandupdatetheirfolders.Encouragestudents to review the inequalities, graphs, and explanations that are needed for the project.

Chapter 3 Project Teacher Notes: Hot! Hot! Hot!


74

Name Class Date

Suppose you are the owner of the Sizzlin’ Sauce Company. Your company makes two different kinds of sauce, Red Hot Sauce and Scorchin’ Hot Sauce.

red Hot sauce

Yield: 1 pint

1 pt tomato sauce with onions5 green peppers, diced4 hot chili peppers, seeded and diced

scorchin’ Hot sauce

Yield: 1 pint

1 pt tomato sauce with onions4 green peppers, diced8 hot chili peppers, seeded and diced

As the owner of a successful business, you want to minimize costs, maximize profit, and keep customers satisfied by filling orders promptly.

List of Materials• Calculator

• Graphpaper

Activities activity 1: GraphingTo fill an order for Sizzlin’ Sauce sauces, you bought 1050 green peppers and 1200 hot chili peppers.

• Writeandgraphasystemofinequalitiestorepresentthenumberofpintsofeach kind of sauce you can make. Refer to the recipes above.

• Selectonesolutionofthesystemanddeterminehowmanypeppersyouwill have left over.

activity 2: analyzingSuppose you make $1.20 profit on 1 pint of Red Hot Sauce and $1.00 profit on 1 pint of Scorchin’ Hot Sauce. Using the restrictions from Activity 1, decide how much of each sauce you should make and sell to maximize your profit. What is the maximum profit?

activity 3: researchingVisit a local grocery store to estimate the cost of each sauce ingredient. Remember that buying in large quantities can save you money.

• Findthecostofproducing1ptofeachtypeofsauce.

• Whatsellingpricewillyousetforeachsaucetomaintainyourprofit?

Chapter 3 Project: Hot! Hot! Hot!

x 5 the number of pints of Red Hot Sauce, y 5 the number of pints of Scorchin’ Hot Sauce; 5x 1 4y " 1050, 4x 1 8y " 1200, x # 0, y # 0; Answers may vary. Sample: If 100 pt of each sauce are produced, there will be 150 green peppers left over.

Red Hot Sauce: 150 pt, Scorchin’ Hot Sauce: 75 pt; $255

Check students’ work.

200

100

00 400200100 300

300

400

Number of Pints ofRed Hot Sauce

Num

ber

of P

ints

of

Scor

chin

' Hot

Sau

ce

Number ofPints of Sauce


75

Name Class Date

Chapter 3 Project: Hot! Hot! Hot! (continued)

activity 4: OrganizingYou can sell your sauce to a supermarket chain, a local grocery, and a specialty store. The supermarket chain will buy 288 pt every 8 weeks, the grocery will buy 60 pt every 4 weeks, and the speciality store will buy 24 pt each week.

• Howmanypintsofsauceshouldyouproduceeachweektofilltheseorders?Presume that you want to produce the same number of pints each week, and that the type of sauce is not a factor in filling these orders.

• Designaspreadsheettotrackyourstockofsauceaftereachweek.Usecellformulas.

Finishing the ProjectThe activities should help you to complete your project. Your report should include your analysis of the cost of producing Sizzlin’ Sauces. Include your profit analysis and production spreadsheet. Illustrate your reasoning and decisions with graphs.

reflect and revisePresent your analysis to a small group of classmates. After you have heard their analyses and presented your own, decide if your work is complete, clear, and convincing. If needed, make changes to improve your presentation.

extending the ProjectAre there other expenses you could expect in addition to those you have already considered? Estimate them. Modify your recommendations if necessary.

Answers may vary. Sample: Spreadsheet with columns A: Week No.; B: Sauce Made (pt); C: Supermarket Orders (pt); D: Local Grocery orders (pt); E: Speciality Store Orders (pt); F: Sauce Available After Orders (pt)

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Chapter 3 Project Manager: Hot! Hot! Hot!

Getting StartedRead the project. As you work on the project, you will need a calculator, graph paper, materials to record your calculations, and materials to make accurate and attractive graphs. Keep all of your work for the project in a folder.

Checklist suggestions

☐ Activity 1: writing and graphing inequalities

☐ Make sure answers are reasonable.

☐ Activity 2: maximizing profit ☐ Check your work with a graphing calculator.

☐ Activity 3: determining selling price ☐ Document your information.

☐ Activity 4: designing production schedule

☐ Remember that you will produce the same amount of sauce every week.

☐ profit analysis ☐ Besides the cost of ingredients, what information would need to be taken into consideration to accurately determine the profit? How would you go about finding this information?

Scoring Rubric4 Correct inequalities are written and graphed. Calculations and graphs are

accurate. The folder is well organized and provides useful information.

3 Minor errors are made. Reasoning and explanations are essentially correct. Graphs contain minor errors in scale or are labeled incorrectly. The folder provides useful information, but needs to be better organized.

2 Inaccurate inequalities are written and graphed. The folder lacks organization. Graphs could be neater and more accurate. Explanations lack detail.

1 Major concepts are misunderstood. Project satisfies few of the requirements and shows poor organization and effort.

0 Major elements of the project are incomplete or missing.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of the Project

ch3 ans

Documents

number of hours

h hours of court use

cost c

campus charges

groups linear systems

solutionsthe system

example linear systems

solution consistent