7and inequalities - wikispaces7... · systems of equations 7 and inequalities ... chapter 7:...

Systems of Equationsand Inequalities77.1 Solve Linear Systems by Graphing

7.2 Solve Linear Systems by Substitution

7.3 Solve Linear Systems by Adding or Subtracting

7.4 Solve Linear Systems by Multiplying First

7.5 Solve Special Types of Linear Systems

7.6 Solve Systems of Linear Inequalities

In previous chapters, you learned the following skills, which you’ll use inChapter 7: graphing linear equations, solving equations, determining whetherlines are parallel, and graphing linear inequalities in two variables.

Prerequisite Skills

VOCABULARY CHECKCopy and complete the statement.

1. The least common multiple of 10 and 15 is ? .

2. Two lines in the same plane are ? if they do not intersect.

SKILLS CHECKGraph the equation. (Review p. 225 for 7.1.)

3. x 2 y 5 4 4. 6x 2 y 5 21 5. 4x 1 5y 5 20 6. 3x 2 2y 5 212

Solve the equation. (Review p. 148 for 7.2–7.4.)

7. 5m 1 4 2 m 5 20 8. 10(z 1 5) 1 z 5 6

Tell whether the graphs of the two equations are parallel lines. Explain yourreasoning. (Review p. 244 for 7.5.)

9. y 5 2x 2 3, y 1 2x 5 23 10. y 2 5x 5 21, y 2 5x 5 1

11. y 5 x 1 10, x 2 y 5 29 12. 6x 2 y 5 4, 4x 2 y 5 6

Graph the inequality. (Review p. 405 for 7.6.)

13. y ≤ 22x 1 1 14. x 2 y < 5 15. x ≥ 24 16. y > 3

Before

424

In Chapter 7, you will apply the big ideas listed below and reviewed in theChapter Summary on page 474. You will also use the key vocabulary listed below.

Big Ideas1 Solving linear systems by graphing

2 Solving linear systems using algebra

3 Solving systems of linear inequalities

• system of linear equations,p. 427

• solution of a system oflinear equations, p. 427

• consistent independentsystem, p. 427

• inconsistent system, p. 459

• consistent dependentsystem, p. 459

• system of linearinequalities, p. 466

• solution of a system oflinear inequalities, p. 466

• graph of a system oflinear inequalities, p. 466

KEY VOCABULARY

You can use a system of linear equations to solve problems about travelingwith and against a current. For example, you can write and solve a system oflinear equations to find the average speed of a kayak in still water.

AlgebraThe animation illustrated below for Example 4 on page 446 helps you answerthis question: What is the average speed of the kayak in still water?

Click the button that will produce anequation in one variable.

You have to find the speed of the kayakin still water.

Algebra at classzone.com

Other animations for Chapter 7: pages 428, 435, 441, 446, 452, 459,and 466

Now

Why?

425

426 Chapter 7 Systems of Equations and Inequalities

7.1 Solving Linear Systems Using Tables MATERIALS • pencil and paper

D R A W C O N C L U S I O N S Use your observations to complete these exercises

1. When Bill and his brother have the same number of books in their collections, how many books will each of them have?

2. Graph the equations above on the same coordinate plane. What do you notice about the graphs and the solution you found above?

Use a table to solve the linear system.

3. y 5 2x 1 3 4. y 5 2x 1 1 5. y 5 23x 1 1y 5 23x 1 18 y 5 2x 2 5 y 5 5x 2 31

x y 5 2x 1 15 y 5 4x 1 7

0 15 7

1 ? ?

2 ? ?

3 ? ?

4 ? ?

5 ? ?

Use before Lesson 7.1

Q U E S T I O N How can you use a table to solve a linear system?

A system of linear equations, or linear system, consists of two or more linear equations in the same variables. A solution of a linear system is an ordered pair that satisfi es each equation in the system. You can use a table to fi nd a solution to a linear system.

E X P L O R E Solve a linear system

Bill and his brother collect comic books. Bill currently has 15 books and adds 2 books to his collection every month. His brother currently has 7 books and adds 4 books to his collection every month. Use the equations below to fi nd the number x of months after which Bill and his brother will have the same number y of comic books in their collections.

y 5 2x 1 15 Number of comic books in Bill’s collection

y 5 4x 1 7 Number of comic books in his brother’s collection

STEP 1 Make a table

Copy and complete the table of values shown.

STEP 2 Find a solution

Find an x-value that gives the same y-value for both equations.

STEP 3 Interpret the solution

Use your answer to Step 2 to find the number of months after which Bill and his brother have the same number of comic books.

ACTIVITYACTIVITYInvestigating Algebra

InvestigatingAlgebr

ggarr

7.1 Solve Linear Systems by Graphing 427

7.1 Before You graphed linear equations.

Now You will graph and solve systems of linear equations.

Why? So you can analyze craft fair sales, as in Ex. 33.

A system of linear equations, or simply a linear system, consists of two ormore linear equations in the same variables. An example is shown below.

x 1 2y 5 7 Equation 1

3x 2 2y 5 5 Equation 2

A solution of a system of linear equations in two variables is an ordered pairthat satisfies each equation in the system.

One way to find the solution of a linear system is by graphing. If the linesintersect in a single point, then the coordinates of the point are the solutionof the linear system. A solution found using graphical methods should bechecked algebraically.

Key Vocabulary• system of linear

equations• solution of a

system of linearequations

• consistentindependentsystem

Solve Linear Systemsby Graphing

E X A M P L E 1 Check the intersection point

Use the graph to solve the system. Thencheck your solution algebraically.



Solution

The lines appear to intersect at the point (3, 2).

CHECK Substitute 3 for x and 2 for y in each equation.

x 1 2y 5 7 3x 2 2y 5 5

3 1 2(2) 0 7 3(3) 2 2(2) 0 5

7 5 7 ✓ 5 5 5 ✓

c Because the ordered pair (3, 2) is a solution of each equation, it is asolution of the system.

x

y

1

2

3x 2 2y 5 5

x 1 2y 5 7

TYPES OF LINEAR SYSTEMS In Example 1, the linear system has exactly onesolution. A linear system that has exactly one solution is called a consistentindependent system because the lines are distinct (are independent) andintersect (are consistent). You will solve consistent independent systems inLessons 7.1–7.4. In Lesson 7.5 you will consider other types of systems.


KEY CONCEPT For Your Notebook

Solving a Linear System Using the Graph-and-Check Method

STEP 1 Graph both equations in the same coordinate plane. Forease of graphing, you may want to write each equation inslope-intercept form.

STEP 2 Estimate the coordinates of the point of intersection.

STEP 3 Check the coordinates algebraically by substituting into eachequation of the original linear system.

E X A M P L E 2 Use the graph-and-check method

Solve the linear system: 2x 1 y 5 27 Equation 1


Solution

STEP 1 Graph both equations.

STEP 2 Estimate the point of intersection. The two lines appear to intersectat (4, 23).

STEP 3 Check whether (4, 23) is a solution by substituting 4 for x and23 for y in each of the original equations.

Equation 1 Equation 2

2x 1 y 5 27 x 1 4y 5 28

2(4) + (23) 0 27 4 1 4(23) 0 28

27 5 27 ✓ 28 5 28 ✓

c Because (4, 23) is a solution of each equation, it is a solution ofthe linear system.

at classzone.com

✓ GUIDED PRACTICE for Examples 1 and 2

Solve the linear system by graphing. Check your solution.

1. 25x 1 y 5 0 2. 2x 1 2y 5 3 3. x 2 y 5 5 5x 1 y 5 10 2x 1 y 5 4 3x 1 y 5 3

x

y

2x 1 y 5 27

x 1 4y 5 28

211


✓ GUIDED PRACTICE for Example 3

4. Solve the linear system in Example 3 to find the number of sessionsafter which the total cost with a season pass, including the cost of thepass, is the same as the total cost without a season pass.

5. WHAT IF? In Example 3, suppose a season pass costs $135. After howmany sessions is the total cost with a season pass, including the cost ofthe pass, the same as the total cost without a season pass?

Solution

Write a system of equations where y is the total cost (in dollars) forx sessions.

EQUATION 1

Totalcost

(dollars)5

Cost persession

(dollars/session)p

Number ofsessions(sessions)

y 5 13 p x

EQUATION 2

Totalcost

(dollars)5

Cost forseason pass

(dollars)1

Cost persession

(dollars/session)p

Number ofsessions(sessions)

y 5 90 1 4 p x

c The correct answer is C. A B C D

★ E X A M P L E 3 Standardized Test Practice

The parks and recreation department in your townoffers a season pass for $90.

• As a season pass holder, you pay $4 per sessionto use the town’s tennis courts.

• Without the season pass, you pay $13 persession to use the tennis courts.

Which system of equations can be used to findthe number x of sessions of tennis after whichthe total cost y with a season pass, includingthe cost of the pass, is the same as the totalcost without a season pass?

A y 5 4x B y 5 4xy 5 13x y 5 90 1 13x

C y 5 13x D y 5 90 1 4x

y 5 90 1 4x y 5 90 1 13x

ELIMINATE CHOICES

You can eliminatechoice A becauseneither of the equationsinclude the cost of aseason pass.



6. WHAT IF? In Example 4, suppose the business has a total of 20 rentalsand collects $420. Find the number of bicycles rented.

1. VOCABULARY Copy and complete: A(n) ? of a system of linearequations in two variables is an ordered pair that satisfies each equationin the system.

2. ★ WRITING Explain how to use the graph-and-check method to solve alinear system of two equations in two variables.

CHECKING SOLUTIONS Tell whether the ordered pair is a solution of thelinear system.

3. (23, 1); 4. (5, 2); 5. (22, 1);x 1 y 5 22 2x 2 3y 5 4 6x 1 5y 5 27x 1 5y 5 2 2x 1 8y 5 11 x 2 2y 5 0

7.1 EXERCISES HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 15 and 31

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 6, 7, 27, 28, 29, and 32

5 MULTIPLE REPRESENTATIONSEx. 35

SKILL PRACTICE

E X A M P L E 4 Solve a multi-step problem

15x 1 30y 5 450

x 1 y 5 25

x

y

5

5

RENTAL BUSINESS A business rents in-line skates andbicycles. During one day, the business has a total of25 rentals and collects $450 for the rentals. Find thenumber of pairs of skates rented and the numberof bicycles rented.

Solution

STEP 1 Write a linear system. Let x be the number of pairs of skatesrented, and let y be the number of bicycles rented.

x 1 y 5 25 Equation for number of rentals

15x 1 30y 5 450 Equation for money collected from rentals

STEP 2 Graph both equations.

STEP 3 Estimate the point of intersection. Thetwo lines appear to intersect at (20, 5).

STEP 4 Check whether (20, 5) is a solution.

20 1 5 0 25 15(20) 1 30(5) 0 450

25 5 25 ✓ 450 5 450 ✓

c The business rented 20 pairs of skates and 5 bicycles.


6. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linear system x 1 y 5 22 and 7x 2 4y 5 8?

A (22, 0) B (0, 22) C (2, 0) D (0, 2)

7. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linearsystem 2x 1 3y 5 12 and 10x 1 3y 5 212?

A (23, 3) B (23, 6) C (3, 3) D (3, 6)

SOLVING SYSTEMS GRAPHICALLY Use the graph to solve the linear system.Check your solution.

8. x 2 y 5 4 9. 2x 1 y 5 22 10. x 1 y 5 5 4x 1 y 5 1 2x 2 y 5 6 22x 1 y 5 24

x

y1

1

x

y

1

1 x

y

1

1

11. ERROR ANALYSIS Describeand correct the error in solvingthe linear system below.

x 2 3y 5 6 Equation 12x 2 3y 5 3 Equation 2

GRAPH-AND-CHECK METHOD Solve the linear system by graphing. Checkyour solution.

12. y 5 2x 1 3 13. y 5 2x 1 4 14. y 5 2x 1 2y 5 x 1 1 y 5 2x 2 8 y 5 4x 1 6

15. x 2 y 5 2 16. x 1 2y 5 1 17. 3x 1 y 5 15x 1 y 5 28 22x 1 y 5 24 y 5 215

18. 2x 2 3y 5 21 19. 6x 1 y 5 37 20. 7x 1 5y 5 235x 1 2y 5 26 4x 1 2y 5 18 29x 1 y 5 211

21. 6x 1 12y 5 26 22. 2x 1 y 5 9 23. 25x 1 3y 5 32x 1 5y 5 0 2x 1 3y 5 15 4x 1 3y 5 30

24. 3}4

x 1 1}4

y 5 13}2

25. 1}5

x 2 2}5

y 5 28}5

26. 21.6x 2 3.2y 5 224

x 2 3}4

y 5 13}2

23}4

x 1 y 5 32.6x 1 2.6y 5 26

27. ★ OPEN – ENDED Find values for m and b so that the system y 5 3}5

x 2 1and y 5 mx 1 b has (5, 2) as a solution.

28. ★ WRITING Solve the linear system shownby graphing. Explain why it is important tocheck your solution.

EXAMPLE 1

on p. 427for Exs. 6–11

EXAMPLE 2

on p. 428for Exs. 12–26

xy

2x – 3y = 32x – 3y = 3

x – 3y = 6x – 3y = 6–1

1

The solution is(3, 21).

y 5 4x 2 1.5 Equation 1y 5 22x 1 1.5 Equation 2

432 ★ 5 STANDARDIZED

TEST PRACTICE5 MULTIPLE

REPRESENTATIONS 5 WORKED-OUT SOLUTIONS

on p. WS1

29. ★ EXTENDED RESPONSE Consider the equation 21}4

x 1 6 5 1}2

x 1 3.

a. Solve the equation using algebra.

b. Solve the linear system below using a graph.

y 5 21}4

x 1 6 Equation 1

y 5 1}2

x 1 3 Equation 2

c. How is the linear system in part (b) related to the original equation?

d. Explain how to use a graph to solve the equation 22}5

x 1 5 5 1}5

x 1 2.

30. CHALLENGE The three lines given below form a triangle. Find thecoordinates of the vertices of the triangle.

Line 1: 23x 1 2y 5 1 Line 2: 2x 1 y 5 11 Line 3: x 1 4y 5 9

EXAMPLES3 and 4

on pp. 429–430for Exs. 31–33

31. TELEVISION The graph shows a projection, from1990 on, of the percent of eighth graders whowatch 1 hour or less of television on a weekdayand the percent of eighth graders who watch morethan 1 hour of television on a weekday. Use thegraph to predict the year when the percent ofeighth graders who watch 1 hour or less will equalthe percent who watch more than 1 hour.

32. ★ MULTIPLE CHOICE A car dealership is offering interest-free car loansfor one day only. During this day, a salesperson at the dealershipsells two cars. One of his clients decides to pay off his $17,424 car in36 monthly payments of $484. His other client decides to pay off his$15,840 car in 48 monthly payments of $330. Which system of equationscan be used to determine the number x of months after which bothclients will have the same loan balance y?

A y 5 2484x B y 5 2484x 1 17,424 y 5 2330x y 5 2330x 1 15,840

C y 5 2484x 1 15,840 D y 5 484x 1 17,424 y 5 2330x 1 17,424 y 5 330x 1 15,840

33. CRAFTS Kirigami is the Japanese art of making paper designsby folding and cutting paper. A student sells small and largegreeting cards decorated with kirigami at a craft fair.The small cards cost $3 per card, and the large cards cost$5 per card. The student collects $95 for sellinga total of 25 cards. How many of each typeof card did the student sell?

PROBLEM SOLVING

Years since 1990Pe

rcen

t of

eigh

th g

rade

rs

20 40 60 80

80

40

00 t

p

1 hour or less

more than 1 hour

7.1 433

34. FITNESS You want to burn 225 calories while exercising at a gym. Thenumber of calories that you burn per minute on different machines atthe gym is shown below.

a. Suppose you have 40 minutes to exercise at the gym and you wantto use the stair machine and stationary bike. How many minutesshould you spend on each machine so that you burn 225 calories?

b. Suppose you have 30 minutes to exercise at the gym and you wantto use the stair machine and the elliptical trainer. How many minutesshould you spend on each machine so that you burn 225 calories?

35. MULTIPLE REPRESENTATIONS It costs $15 for a yearly membership toa movie club at a movie theater. A movie ticket costs $5 for club membersand $8 for nonmembers.

a. Writing a System of Equations Write a system of equations that youcan use to find the number x of movies viewed after which the total

cost y for a club member, including the membership fee, is the sameas the cost for a nonmember.

b. Making a Table Make a table of values that shows the total cost for aclub member and a nonmember after paying to see 1, 2, 3, 4, 5, and

6 movies.

c. Drawing a Graph Use the table to graph the system of equations.Under what circumstances does it make sense to become a movieclub member? Explain your answer by using the graph.

36. CHALLENGE With a minimum purchase of $25, you can open a creditaccount with a clothing store. The store is offering either $25 or 20% offof your purchase if you open a credit account. You decide to open a creditaccount. Should you choose $25 or 20% off of your purchase? Explain.

Stair machine Elliptical trainer Stationary bike

You burn 5 Cal/min. You burn 8 Cal/min. You burn 6 Cal/min.

EXTRA PRACTICE for Lesson 7.1, p. 944 ONLINE QUIZ at classzone.com

PREVIEW

Prepare forLesson 7.2in Exs. 43–48.

Solve the equation.

37. x 1 5 5 214 (p. 134) 38. 25x 1 6 5 21 (p. 141)

39. 3(x 1 2) 5 26 (p. 148) 40. 11x 1 9 5 13x 2 3 (p. 154)

41. 3x 2 8 5 11x 1 12 (p. 154) 42. 4(x 1 1) 5 22x 2 18 (p. 154)

Write the equation so that y is a function of x. (p. 184)

43. 5y 1 25 5 3x 44. 7x 5 y 1 9 45. 4y 1 11 5 3y 1 4x

46. 6x 1 3y 5 2x 1 3 47. y 1 2x 1 6 5 21 48. 4x 2 12 5 5x 1 2y

MIXED REVIEW


7.1 Solving Linear Systemsby Graphing

P R A C T I C E

Solve the linear system using a graphing calculator.

1. y 5 x 1 4 2. 5x 1 y 5 24 3. 20.45x 2 y 5 1.35 4. 20.4x 1 0.8y 5 216y 5 23x 2 2 x 2 y 5 22 21.8x 1 y 5 21.8 1.2x 1 0.4y 5 1

STEP 3 Display graph

Graph the equations using a standardviewing window.

STEP 4 Find point of intersection

Use the intersect feature to find thepoint where the graphs intersect.

The solution is about (0.47, 1.8).

Y1=-(5/2)X+3Y2=(1/3)X+(5/3)Y3=Y4=Y5=Y6=Y7=

X=.47058824 Y=1.8235294

Q U E S T I O N How can you use a graphing calculator to solve alinear system?

E X A M P L E Solve a linear system

Solve the linear system using a graphing calculator.



Use after Lesson 7.1 Graphing

Calculator ACTIVITYAACTIVITYGraphingCalculator

p gp


5x 1 2y 5 6 x 2 3y 5 25

2y 5 25x 1 6 23y 5 2x 2 5

y 5 25}2x 1 3 y 5 1

}3x 1 5

}3

STEP 1 Rewrite equations

Solve each equation for y.

STEP 2 Enter equations

Press and enter the equations.

classzone.com Keystrokes

7.2 Solve Linear Systems by Substitution 435

7.2 Solve Linear Systemsby Substitution


equations, p. 427


Solving a Linear System Using the Substitution Method

STEP 1 Solve one of the equations for one of its variables. When possible,solve for a variable that has a coefficient of 1 or 21.

STEP 2 Substitute the expression from Step 1 into the other equation andsolve for the other variable.

STEP 3 Substitute the value from Step 2 into the revised equation fromStep 1 and solve.

Solve the linear system: y 5 3x 1 2 Equation 1x 1 2y 5 11 Equation 2

Solution

STEP 1 Solve for y. Equation 1 is already solved for y.

STEP 2 Substitute 3x 1 2 for y in Equation 2 and solve for x.

x 1 2y 5 11 Write Equation 2.

x 1 2(3x 1 2) 5 11 Substitute 3x 1 2 for y.

7x 1 4 5 11 Simplify.

7x 5 7 Subtract 4 from each side.

x 5 1 Divide each side by 7.

STEP 3 Substitute 1 for x in the original Equation 1 to find the value of y.

y 5 3x 1 2 5 3(1) 1 2 5 3 1 2 5 5

c The solution is (1, 5).

CHECK Substitute 1 for x and 5 for y in each of the original equations.

y 5 3x 1 2 x 1 2y 5 11

5 0 3(1) 1 2 1 1 2(5) 0 11

5 5 5 ✓ 11 5 11 ✓

E X A M P L E 1 Use the substitution method

Before You solved systems of linear equations by graphing.

Now You will solve systems of linear equations by substitution.

Why? So you can find tubing costs, as in Ex. 32.

at classzone.com


E X A M P L E 2 Use the substitution method

Solve the linear system: x 2 2y 5 26 Equation 14x 1 6y 5 4 Equation 2

Solution

STEP 1 Solve Equation 1 for x.

x 2 2y 5 26 Write original Equation 1.

x 5 2y 2 6 Revised Equation 1

STEP 2 Substitute 2y 2 6 for x in Equation 2 and solve for y.

4x 1 6y 5 4 Write Equation 2.

4(2y 2 6) 1 6y 5 4 Substitute 2y – 6 for x.

8y 2 24 1 6y 5 4 Distributive property

14y 2 24 5 4 Simplify.

14y 5 28 Add 24 to each side.

y 5 2 Divide each side by 14.

STEP 3 Substitute 2 for y in the revised Equation 1 to find the value of x.

x 5 2y 2 6 Revised Equation 1

x 5 2(2) 2 6 Substitute 2 for y.

x 5 22 Simplify.




x 2 2y 5 26 4x 1 6y 5 4

22 2 2(2) 0 26 4(22) 1 6(2) 0 4

26 5 26 ✓ 4 5 4 ✓


Solve the linear system using the substitution method.

1. y 5 2x 1 5 2. x 2 y 5 3 3. 3x 1 y 5 273x 1 y 5 10 x 1 2y 5 26 22x 1 4y 5 0

x

y

4x 1 6y 5 4

x 2 2y 5 262

21

CHECK REASONABLENESS When solvinga linear system using the substitutionmethod, you can use a graph to checkthe reasonableness of your solution. Forexample, the graph at the right verifies that(22, 2) is a solution of the linear system inExample 2.

CHOOSE ANEQUATION

Equation 1 was chosenin Step 1 because x hasa coefficient of 1. So,only one step is neededto solve Equation 1for x.


E X A M P L E 3 Solve a multi-step problem


4. In Example 3, what is the total cost for website hosting for each companyafter 20 months?

5. WHAT IF? In Example 3, suppose the Internet service provider offers$5 off the set-up fee. After how many months will the total cost for websitehosting be the same for both companies?

ANOTHER WAY

For an alternativemethod for solving theproblem in Example 3,turn to page 442 forthe Problem SolvingWorkshop.

WEBSITES Many businesses pay website hosting companies to store andmaintain the computer files that make up their websites. Internet serviceproviders also offer website hosting. The costs for website hosting offered bya website hosting company and an Internet service provider are shown inthe table. Find the number of months after which the total cost for websitehosting will be the same for both companies.

Company Set-up fee(dollars)

Cost per month(dollars)

Internet service provider 10 21.95

Website hosting company None 22.45

Solution

STEP 1 Write a system of equations. Let y be the total cost after x months.

Equation 1: Internet service provider

Totalcost 5

Set-upfee 1

Cost permonth p

Numberof months

y 5 10 1 21.95 p x

Equation 2: Website hosting company

Totalcost 5

Cost permonth p

Numberof months

y 5 22.45 p x

The system of equations is: y 5 10 1 21.95x Equation 1

y 5 22.45x Equation 2

STEP 2 Substitute 22.45x for y in Equation 1 and solve for x.

y 5 10 1 21.95x Write Equation 1.

22.45x 5 10 1 21.95x Substitute 22.45x for y.

0.5x 5 10 Subtract 21.95x from each side.

x 5 20 Divide each side by 0.5.

c The total cost will be the same for both companies after 20 months.


E X A M P L E 4 Solve a mixture problem

ANTIFREEZE For extremely cold temperatures, an automobile manufacturerrecommends that a 70% antifreeze and 30% water mix be used in thecooling system of a car. How many quarts of pure (100%) antifreeze and a50% antifreeze and 50% water mix should be combined to make 11 quarts ofa 70% antifreeze and 30% water mix?

Solution

STEP 1 Write an equation for the total number of quarts and an equationfor the number of quarts of antifreeze. Let x be the number ofquarts of 100% antifreeze, and let y be the number of quarts a50% antifreeze and 50% water mix.

Equation 1: Total number of quartsx 1 y 5 11

Equation 2: Number of quarts of antifreeze

x quarts of100% antifreeze

y quarts of50%–50% mix

11 quarts of70%–30% mix

100% 150%

570%

1 p x 1 0.5 p y 5 0.7(11)

x 1 0.5y 5 7.7

The system of equations is: x 1 y 5 11 Equation 1

x 1 0.5y 5 7.7 Equation 2

STEP 2 Solve Equation 1 for x.

x 1 y 5 11 Write Equation 1.

x 5 11 2 y Revised Equation 1

STEP 3 Substitute 11 2 y for x in Equation 2 and solve for y.

x 1 0.5y 5 7.7 Write Equation 2.

(11 – y) 1 0.5y 5 7.7 Substitute 11 2 y for x.

y 5 6.6 Solve for y.

STEP 4 Substitute 6.6 for y in the revised Equation 1 to find the value of x.

x 5 11 2 y 5 11 2 6.6 5 4.4

c Mix 4.4 quarts of 100% antifreeze and 6.6 quarts of a 50% antifreeze and50% water mix to get 11 quarts of a 70% antifreeze and 30% water mix.


6. WHAT IF? How many quarts of 100% antifreeze and a 50% antifreeze and50% water mix should be combined to make 16 quarts of a 70% antifreezeand 30% water mix?

DRAW A DIAGRAM

Each bar shows theliquid in each mix. Thegreen portion shows thepercent of the mix thatis antifreeze.


1. VOCABULARY Give an example of a system of linear equations.

2. ★ WRITING If you are solving the linear systemshown using the substitution method, whichequation would you solve for which variable?Explain.

SOLVING LINEAR SYSTEMS Solve the linear system using substitution.

3. x 5 17 2 4y 4. y 5 2x 2 1 5. x 5 y 1 3y 5 x 2 2 2x 1 y 5 3 2x 2 y 5 5

6. 4x 2 7y 5 10 7. x 5 16 2 4y 8. 25x 1 3y 5 51y 5 x 2 7 3x 1 4y 5 8 y 5 10x 2 8

9. 2x 5 12 10. 2x 2 y 5 23 11. x 1 y 5 0x 2 5y 5 229 x 2 9 5 21 x 2 2y 5 6

12. 2x 1 y 5 9 13. 5x 1 2y 5 9 14. 5x 1 4y 5 324x 2 y 5 215 x 1 y 5 23 9x 2 y 5 33

15. 11x 2 7y 5 214 16. 20x 2 30y 5 250 17. 6x 1 y 5 4x 2 2y 5 24 x 1 2y 5 1 x 2 4y 5 19

18. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linear system4x 2 y 5 17 and 29x 1 8y 5 2?

A (6, 7) B (7, 6) C (7, 11) D (11, 7)

19. ERROR ANALYSIS Describe and correct the error in solving the linearsystem 4x 1 2y 5 6 and 3x 1 y 5 9.

SOLVING LINEAR SYSTEMS Solve the linear system using substitution.

20. 4.5x 1 1.5y 5 24 21. 35x 1 y 5 20 22. 3x 2 2y 5 8x 2 y 5 4 1.5x 2 0.1y 5 18 0.5x 1 y 5 17

23. 0.5x 1 0.6y 5 5.7 24. x 2 9 5 0.5y 25. 0.2x 1 y 5 21.8

2x 2 y 5 21 2.2x 2 3.1y 5 20.2 1.8y 1 5.5x 5 27.6

26. 1}2

x 1 1}4

y 5 5 27. x 1 1}3

y 5 22 28. 3}8

x 1 3}4

y 5 12

x 2 1}2

y 5 1 28x 22}3y 5 4 2

}3

x 1 1}2

y 5 13

7.2 EXERCISES


2x 1 y 5 8 Equation 2

EXAMPLE 1


EXAMPLE 2


Step 1 Step 2 Step 3 The solution3x 1 y 5 9 4x 1 2(9 2 3x) 5 6 y 5 9 2 3x is (6, 1).

y 5 9 2 3x 4x 1 18 2 6x 5 6 6 5 9 2 3x 22x 5 212 23 5 23x x 5 6 1 5 x

HOMEWORKKEY


★ 5 STANDARDIZED TEST PRACTICEExs. 2, 18, 29, 33, and 37

SKILL PRACTICE

440

29. ★ WRITING Suppose you solve a linear system using substitution.Explain how you can use a graph to check your solution.

30. CHALLENGE Find values of a and b so that thelinear system shown has a solution of (29, 4).

31. FUNDRAISING During a football game, the parents of the football playerssell pretzels and popcorn to raise money for new uniforms. They charge$2.50 for a bag of popcorn and $2 for a pretzel. The parents collect $336in sales during the game. They sell twice as many bags of popcorn aspretzels. How many bags of popcorn do they sell? How many pretzels dothey sell?

32. TUBING COSTS The members of an outing club take a day-long tubing tripdown a river. The company that offers the tubing trip charges $15 to rent atube for a person to use and $7.50 to rent a “cooler” tube, which is used tocarry food and water in a cooler. The club members spend $360 to rent atotal of 26 tubes. How many of each type of tube do they rent?

33. ★ SHORT RESPONSE In the mobile shown, objects areattached to each end of a dowel. For the dowel to balance,the following must be true:

x pWeight hanging

from point A 5 y pWeight hanging

from point B

The weight of the objects hanging from point A is 1.5 pounds,and the weight of the objects hanging from point B is1.2 pounds. The length of the dowel is 9 inches. How far frompoint A should the string be placed? Explain.

34. MULTI-STEP PROBLEM Two swimming teams are competing in a400 meter medley relay. During the last leg of the race, the swimmer inlane 1 has a 1.2 second head start on the swimmer in lane 2, as shown.

a. Let t be the time since the swimmer in lane 2 started the last leg. Afterhow many seconds into the leg will the swimmer in lane 2 catch up tothe swimmer in lane 1?

b. Does the swimmer in lane 2 catch up to the swimmer in lane 1 beforethe race ends? Explain.

PROBLEM SOLVING

★ 5 STANDARDIZEDTEST PRACTICE

5 WORKED-OUT SOLUTIONSon p. WS1

ax 1 by 5 216 Equation 1


EXAMPLE 3

on p. 437for Exs. 31–33

1

2

Swimming at 1.8 m/secwith a 1.2 sec head start

Swimming at 1.9 m/sec

441

35. CHEMISTRY In your chemistry lab, you have a bottle of 1% hydrochloricacid solution and a bottle of 5% hydrochloric acid solution. You need100 milliliters of a 3% hydrochloric acid solution for an experiment. Howmany milliliters of each solution do you need to mix together?

36. MONEY Laura has $4.50 in dimes and quarters. She has 3 more dimesthan quarters. How many quarters does she have?

37. ★ SHORT RESPONSE A gazelle can run 73 feet per second for severalminutes. A cheetah can run 88 feet per second, but it can sustain thisspeed for only 20 seconds. A gazelle is 350 feet from a cheetah whenboth animals start running. Can the gazelle stay ahead of the cheetah?Explain.

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38. CHALLENGE A gardener needs 6 bushels of a potting medium of 40% peatmoss and 60% vermiculite. He decides to add 100% vermiculite to hiscurrent potting medium that is 50% peat moss and 50% vermiculite. Thegardener has 5 bushels of the 50% peat moss and 50% vermiculite mix.Does he have enough of the 50% peat moss and 50% vermiculite mix tomake 6 bushels of the 40% peat moss and 60% vermiculite mix? Explain.

QUIZ for Lessons 7.1–7.2

Solve the linear system by graphing. Check your solution. (p. 427)

1. x 1 y 5 22 2. x 2 y 5 0 3. x 2 2y 5 122x 1 y 5 6 5x 1 2y 5 27 23x 1 y 5 21

Solve the linear system using substitution. (p. 435)

4. y 5 x 2 4 5. y 5 4 2 3x 6. x 5 y 1 922x 1 y 5 18 5x 2 y 5 22 5x 2 3y 5 7

7. 2y 1 x 5 24 8. 5x 2 4y 5 27 9. 3x 2 5y 5 13y 2 x 5 25 22x 1 y 5 3 x 1 4y 5 10


EXAMPLE 4

on p. 438for Ex. 35

Solve the proportion. Check your solution.

39. 3}7

5 x}21

(p. 162) 40.y

}30

5 9}10

(p. 162) 41. 12}16

5 3z}4

(p. 162)

42. 35}q 5 5

}3

(p. 168) 43. 3}2r

5 26}16

(p. 168) 44. 4}3

5 s}s 2 2

(p. 168)

Write two equations in standard form that are equivalent to thegiven equation. (p. 311)

45. x 2 4y 5 0 46. 23x 1 9y 5 6 47. 27x 2 y 5 1

48. 5x 2 10y 5 5 49. 22x 2 12y 5 8 50. 6x 1 15y 5 23

MIXED REVIEW

PREVIEW

Prepare forLesson 7.3in Exs. 45–50.


LESSON 7.2Another Way to Solve Example 3, page 437

MULTIPLE REPRESENTATIONS In Example 3 on page 437, you saw howto solve the problem about website hosting by solving a linear systemalgebraically. You can also solve the problem using a table.

WEBSITES Many businesses pay website hosting companies to storeand maintain the computer files that make up their websites. Internetservice providers also offer website hosting. The costs for websitehosting offered by a website hosting company and an Internet serviceprovider are shown in the table. Find the number of months after whichthe total cost for website hosting will be the same for both companies.

Company Set-up fee Cost per month

Internet service provider $10 $21.95

Website hosting company None $22.45

PR AC T I C E

1. TAXIS A taxi company charges $2.80 for thefirst mile and $1.60 for each additional mile.Another taxi company charges $3.20 for thefirst mile and $1.50 for each additional mile.After how many miles will each taxi cost thesame? Use a table to solve the problem.

2. SCHOOL PLAY An adult ticket to a schoolplay costs $5 and a student ticket costs $3. Atotal of $460 was collected from the sale of120 tickets. How many student tickets werepurchased? Solve the problem using algebra.Then use a table to check your answer.

PRO B L E M

M E T H O D Making a Table An alternative approach is to make a table.

STEP 1 Make a table for the totalcost of website hosting forboth companies.

Include the set-up fee in thecost for the first month.

STEP 2 Look for the month inwhich the total cost of theservice from the Internetservice provider and thewebsite hosting companyis the same. This happensafter 20 months.

MonthsInternetservice

provider

Websitehosting

company

1 $31.95 $22.45

2 $53.90 $44.90

3 $75.85 $67.35

A A A

19 $427.05 $426.55

20 $449.00 $449.00

21 $470.95 $471.45

ALTERNATIVE METHODSALTERNATIVE METHODSUsingUsing

7.3 Solve Linear Systems by Adding or Subtracting 443

E X P L O R E Solve a linear system using algebra tiles.

Solve the linear system: 3x 2 y 5 5 Equation 1x 1 y 5 3 Equation 2

STEP 1 Model equations

Model each equation using algebra tiles. Arrange the algebra tiles so that one equation is directly below the other equation.

STEP 2 Add equations

Combine the two equations to form one equation. Notice that the new equation has one positive y-tile and one negative y-tile. The y-tiles can be removed because the pair of y-tiles has a value of 0.

STEP 3 Solve for x

Divide the remaining tiles into four equal groups. Each x-tile is equal to two 1-tiles. So, x 5 2.

STEP 4 Solve for y

To find the value of y, use the model for Equation 2. Because x 5 2, you can replace the x-tile with two 1-tiles. Solve the new equation for y. So y 5 1, and the solution of the system is (2, 1).

7.3 Linear Systems and Elimination MATERIALS • algebra tiles

D R A W C O N C L U S I O N S Use your observations to complete these exercises

Use algebra tiles to model and solve the linear system.

1. x 1 3y 5 8 2. 2x 1 y 5 5 3. 5x 2 2y 5 22 4. x 1 2y 5 34x 2 3y 5 2 22x 1 3y 5 7 x 1 2y 5 14 2x 1 3y 5 2

5. REASONING Is it possible to solve the linear system 3x 2 2y 5 6 and 2x 1 y 5 11 using the steps shown above? Explain your reasoning.

1 2 1 2 1 2

2111 51 11 1

1

1 5 1 1 11

1111

51 1 1

1 11 1

1

12

1 1 11

51 1

1 1 11 1 1

Use before Lesson 7.3ACTIVITYACTIVITYInvestigating Algebra

InvestigatingAlgebra

g gg

5 1 1 11 1 1

Q U E S T I O N How can you solve a linear system using algebra tiles?

You can use the following algebra tiles to model equations.

1-tiles x-tiles y-tiles


7.3


equations, p. 427

When solving a linear system, you can sometimes add or subtract theequations to obtain a new equation in one variable. This method is calledelimination.

E X A M P L E 1 Use addition to eliminate a variable

Solve the linear system: 2x 1 3y 5 11 Equation 122x 1 5y 5 13 Equation 2

Solution

STEP 1 Add the equations toeliminate one variable.

STEP 2 Solve for y.

STEP 3 Substitute 3 for y in either equation and solve for x.


2x 1 3(3) 5 11 Substitute 3 for y.

x 5 1 Solve for x.



2x 1 3y 5 11 22x 1 5y 5 13

2(1) 1 3(3) 0 11 22(1) 1 5(3) 0 13

11 5 11 ✓ 13 5 13 ✓


Solving a Linear System Using the Elimination Method

STEP 1 Add or subtract the equations to eliminate one variable.

STEP 2 Solve the resulting equation for the other variable.

STEP 3 Substitute in either original equation to find the value of theeliminated variable.

2x 1 3y 5 11 22x 1 5y 5 13

8y 5 24

y 5 3

Solve Linear Systems byAdding or Subtracting

ADD EQUATIONS

When the coefficientsof one variable areopposites, add theequations to eliminatethe variable.

Before You solved linear systems by graphing and using substitution.

Now You will solve linear systems using elimination.

Why? So you can solve a problem about arranging flowers, as in Ex. 42.


E X A M P L E 2 Use subtraction to eliminate a variable

Solve the linear system: 4x 1 3y 5 2 Equation 15x 1 3y 5 22 Equation 2

Solution

STEP 1 Subtract the equations toeliminate one variable.

STEP 2 Solve for x.

STEP 3 Substitute 24 for x in either equation and solve for y.


4(24) 1 3y 5 2 Substitute 24 for x.

y 5 6 Solve for y.


✓ GUIDED PRACTICE for Examples 1, 2, and 3

Solve the linear system.

1. 4x 2 3y 5 5 2. 25x 2 6y 5 8 3. 6x 2 4y 5 1422x 1 3y 5 27 5x 1 2y 5 4 23x 1 4y 5 1

4. 7x 2 2y 5 5 5. 3x 1 4y 5 26 6. 2x 1 5y 5 127x 2 3y 5 4 2y 5 3x 1 6 5y 5 4x 1 6

4x 1 3y 5 25x 1 3y 5 22

2x 5 4

x 5 24

E X A M P L E 3 Arrange like terms

Solve the linear system: 8x 2 4y 5 24 Equation 14y 5 3x 1 14 Equation 2

Solution

STEP 1 Rewrite Equation 2 so that the like terms are arranged in columns.

STEP 2 Add the equations.

STEP 3 Solve for x.

STEP 4 Substitute 2 for x in either equation and solve for y.

4y 5 3x 1 14 Write Equation 2.

4y 5 3(2) 1 14 Substitute 2 for x.

y 5 5 Solve for y.


AVOID ERRORS

Make sure that theequal signs are in thesame column, just asthe like terms are.

8x 2 4y 5 244y 5 3x 1 14

8x 2 4y 5 24 23x 1 4y 5 14

5x 5 10

x 5 2

SUBTRACTEQUATIONS

When the coefficientsof one variable arethe same, subtract theequations to eliminatethe variable.


E X A M P L E 4 Write and solve a linear system

KAYAKING During a kayaking trip, a kayaker travels 12 miles upstream(against the current) and 12 miles downstream (with the current), as shown.The speed of the current remained constant during the trip. Find the averagespeed of the kayak in still water and the speed of the current.

STEP 1 Write a system of equations. First find the speed of the kayakgoing upstream and the speed of the kayak going downstream.

Upstream: d 5 rt Downstream: d 5 rt

12 5 r p 3 12 5 r p 2

4 5 r 6 5 r

Use the speeds to write a linear system. Let x be the average speedof the kayak in still water, and let y be the speed of the current.

Equation 1: Going upstream

Speed of kayakin still water 2

Speed ofcurrent 5

Speed of kayakgoing upstream

x 2 y 5 4

Equation 2: Going downstream

Speed of kayakin still water 1

Speed ofcurrent 5

Speed of kayakgoing downstream

x 1 y 5 6

STEP 2 Solve the system of equations.

x 2 y 5 4 Write Equation 1.x 1 y 5 6 Write Equation 2.

2x 5 10 Add equations.

x 5 5 Solve for x.

Substitute 5 for x in Equation 2 and solve for y.

5 1 y 5 6 Substitute 5 for x in Equation 2.

y 5 1 Subtract 5 from each side.

c The average speed of the kayak in still water is 5 miles per hour, and thespeed of the current is 1 mile per hour.

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COMBINE SPEEDS

When you go upstream,the speed at whichyou can travel in stillwater is decreasedby the speed of thecurrent. The oppositeis true when you godownstream.

DIRECTION OF CURRENT

Upstream: 3 hours

Downstream: 2 hours



7. WHAT IF? In Example 4, suppose it takes the kayaker 5 hours to travel10 miles upstream and 2 hours to travel 10 miles downstream. The speedof the current remains constant during the trip. Find the average speed ofthe kayak in still water and the speed of the current.

1. VOCABULARY Give an example of a linear system in two variables thatcan be solved by first adding the equations to eliminate one variable.

2. ★ WRITING Explain how to solvethe linear system shown using theelimination method.

USING ADDITION Solve the linear system using elimination.

3. x 1 2y 5 13 4. 9x 1 y 5 2 5. 23x 2 y 5 82x 1 y 5 5 24x 2 y 5 217 7x 1 y 5 212

6. 3x 2 y 5 30 7. 29x 1 4y 5 217 8. 23x 2 5y 5 2723x 1 7y 5 6 9x 2 6y 5 3 24x 1 5y 5 14

USING SUBTRACTION Solve the linear system using elimination.

9. x 1 y 5 1 10. x 2 y 5 24 11. 2x 2 y 5 722x 1 y 5 4 x 1 3y 5 4 2x 1 7y 5 31

12. 6x 1 y 5 210 13. 5x 1 6y 5 50 14. 4x 2 9y 5 2215x 1 y 5 210 2x 1 6y 5 26 4x 1 3y 5 29

15. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linearsystem 4x 1 9y 5 22 and 11x 1 9y 5 26?

A (22, 4) B (2, 24) C (4, 22) D (4, 2)

ARRANGING LIKE TERMS Solve the linear system using elimination.

16. 2x 2 y 5 32 17. 28y 1 6x 5 36 18. 2x 2 y 5 211y 2 5x 5 13 6x 2 y 5 15 y 5 22x 2 13

19. 2x 2 y 5 14 20. 11y 2 3x 5 18 21. 25x 1 y 5 223x 5 5y 2 38 23x 5 216y 1 33 2y 5 3x 2 9

22. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linearsystem 2x 1 y 5 10 and 3y 5 2x 1 6?

A (23, 24) B (3, 4) C (24, 3) D (4, 3)

7.3 EXERCISES

EXAMPLE 2


EXAMPLE 1


EXAMPLE 3

on p. 445for Exs. 16–22

2x 2 y 5 2 Equation 1


HOMEWORKKEY




SKILL PRACTICE

448

ERROR ANALYSIS Describe and correct the error in finding the value of oneof the variables in the given linear system.

23. 5x 2 7y 5 16 24. 3x 2 2y 5 232x 2 7y 5 8 5y 5 60 2 3x

5x 2 7y 5 162x 2 7y 5 8

4x 5 24x 5 6

3x 2 2y 5 2323x 1 5y 5 60

3y 5 57y 5 19

SOLVING LINEAR SYSTEMS Solve the linear system using elimination.

25. 2x 1 1}2

y 5 219 26. 1}4

x 2 2}3

y 5 7 27. 8x 2 1}2

y 5 238

x 2 y 5 12 1}2

x 1 2}3

y 5 24 1}4

x 2 1}2

y 5 27

28. 5.2x 1 3.5y 5 54 29. 1.3x 2 3y 5 217.6 30. 22.6x 2 3.2y 5 4.823.6x 1 3.5y 5 10 21.3x 1 4.5y 5 25.1 1.9x 2 3.2y 5 24.2

31. 4}5

x 1 2}5

y 5 14 32. 2.7x 1 1.5y 5 36 33. 4 2 4.8x 5 1.7y

2}5

y 1 1}5

x 5 113.5y 5 2.7x 2 6 12.8 1 1.7y 5 213.2x

34. WRITING AN EQUATION OF A LINE Use the following steps to write anequation of the line that passes through the points (1, 2) and (24, 12).

a. Write a system of linear equations by substituting 1 for x and 2 for yin y 5 mx 1 b and 24 for x and 12 for y in y 5 mx 1 b.

b. Solve the system of linear equations from part (a). What is the slopeof the line? What is the y-intercept?

c. Write an equation of the line that passes through (1, 2) and (24, 12).

35. GEOMETRY The rectangle has a perimeterP of 14 feet, and twice its length l is equal to 1less than 4 times its width w. Write and solvea system of linear equations to find the lengthand the width of the rectangle.

36. ★ SHORT RESPONSE Find the solution of the system of linear equationsbelow. Explain your steps.

x 1 3y 5 8 Equation 1 x 2 6y 5 219 Equation 2 5x 2 3y 5 214 Equation 3

37. CHALLENGE For a Þ 0, what is the solution of the system ax 1 2y 5 4and ax 2 3y 5 26?

38. CHALLENGE Solve for x, y, and z in the system of equations below.Explain your steps.

x 1 7y 1 3z 5 29 Equation 13z 1 x 2 2y 5 27 Equation 2

5y 5 10 2 2x Equation 3

l

wP 5 14 ft


5 MULTIPLEREPRESENTATIONS



PROBLEM SOLVING

39. ROWING During a practice, a 4 person crew team rows a rowing shellupstream (against the current) and then rows the same distancedownstream (with the current). The shell moves upstream at a speedof 4.3 meters per second and downstream at a speed of 4.9 meters persecond. The speed of the current remains constant. Use the modelsbelow to write and solve a system of equations to find the average speedof the shell in still water and the speed of the current.

Upstream

Speed of shellin still water 2

Speed ofcurrent 5

Speedof shell

Downstream

Speed of shellin still water 1

Speed ofcurrent 5

Speedof shell

40. OIL CHANGE Two cars get an oil change at the same service center. Eachcustomer is charged a fee x (in dollars) for the oil change plus y dollarsper quart of oil used. The oil change for the car that requires 5 quarts ofoil costs $22.45. The oil change for the car that requires 7 quarts of oilcosts $25.45. Find the fee and the cost per quart of oil.

41. PHONES Cellular phone ring tones can be monophonic or polyphonic.Monophonic ring tones play one tone at a time, and polyphonic ringtones play multiple tones at a time. The table shows the ring tonesdownloaded from a website by two customers. Use the informationto find the cost of a monophonic ring tone and a polyphonic ringtone, assuming that all monophonic ring tones cost the same and allpolyphonic ring tones cost the same.

42. MULTIPLE REPRESENTATIONS For a floral arrangement class, Aliciahas to create an arrangement of twigs and flowers that has a total of9 objects. She has to pay for the twigs and flowers that she uses in herarrangement. Each twig costs $1, and each flower costs $3.

a. Writing a System Alicia spends $15 on the twigs and flowers. Writeand solve a linear system to find the number of twigs and the numberof flowers she used.

b. Making a Table Make a table showing the number of twigs in thearrangement and the total cost of the arrangement when the numberof flowers purchased is 0, 1, 2, 3, 4, or 5. Use the table to check youranswer to part (a).

EXAMPLE 4

on p. 446for Exs. 39–41

CustomerMonophonic

ring tonesPolyphonicring tones

Total cost(dollars)

Julie 3 2 12.85

Tate 1 2 8.95


43. MULTI-STEP PROBLEM On a typical day with light winds, the 1800 mileflight from Charlotte, North Carolina, to Phoenix, Arizona, takes longerthan the return trip because the plane has to fly into the wind.

a. The flight from Charlotte to Phoenix is 4 hours 30 minutes long, andthe flight from Phoenix to Charlotte is 4 hours long. Find the averagespeed (in miles per hour) of the airplane on the way to Phoenix andon the return trip to Charlotte.

b. Let s be the speed (in miles per hour) of the plane with no wind, andlet w be the speed (in miles per hour) of the wind. Use your answerto part (a) to write and solve a system of equations to find the speedof the plane with no wind and the speed of the wind.

44. ★ SHORT RESPONSE The students in the graduating classes at the threehigh schools in a school district have to pay for their caps and gowns.A cap-and-gown set costs x dollars, and an extra tassel costs y dollars.At one high school, students pay $3262 for 215 cap-and-gown sets and72 extra tassels. At another high school, students pay $3346 for 221 cap-and-gown sets and 72 extra tassels. How much will students at the thirdhigh school pay for 218 cap-and-gown sets and 56 extra tassels? Explain.

45. CHALLENGE A clothing manufacturer makes men’s dress shirts. For theproduction process, an ideal sleeve length x (in centimeters) for eachshirt size and an allowable deviation y (in centimeters) from the ideallength are established. The deviation is expressed as 6y. For a specificshirt size, the minimum allowable sleeve length is 62.2 centimeters andthe maximum allowable sleeve length is 64.8 centimeters. Find the idealsleeve length and the allowable deviation.


MIXED REVIEW

Graph the equation.

46. x 2 5y 5 212 (p. 225) 47. 22x 1 3y 5 215 (p. 225)

48. y 1 9 5 2(x 1 2) (p. 302) 49. y 2 4 5 2}3

(x 1 1) (p. 302)

Solve the linear system by graphing. Check your solution. (p. 427)

50. y 5 x 2 3 51. y 5 2 52. 2x 1 y 5 6y 5 2x 1 1 2x 2 3y 5 6 6x 2 2y 5 212

Find the least common multiple of the pair of numbers. (p. 910)

53. 9, 12 54. 18, 24 55. 15, 20

PREVIEW

Prepare forLesson 7.4 inExs. 53–55.

7.4 Solve Linear Systems by Multiplying First 451

7.4

In a linear system like the one below, neither variable can be eliminated byadding or subtracting the equations. For systems like these, you can multiplyone or both of the equations by a constant so that adding or subtracting theequations will eliminate one variable.

5x 1 2y 5 16 10x 1 4y 5 32

3x 2 4y 5 20 3x 2 4y 5 20

Solve Linear Systemsby Multiplying First

Key Vocabulary• least common

multiple, p. 910

The new system isequivalent to theoriginal system.

3 2

Solve the linear system: 6x 1 5y 5 19 Equation 1 2x 1 3y 5 5 Equation 2

Solution

STEP 1 Multiply Equation 2 by 23 so that the coefficients of x are opposites.

6x 1 5y 5 19 6x 1 5y 5 192x 1 3y 5 5 26x 2 9y 5 215

STEP 2 Add the equations. 24y 5 4

STEP 3 Solve for y. y 5 21

STEP 4 Substitute 21 for y in either of the original equations and solvefor x.


2x 1 3(21) 5 5 Substitute 21 for y.

2x 1 (23) 5 5 Multiply.

2x 5 8 Subtract 23 from each side.

x 5 4 Divide each side by 2.




6x 1 5y 5 19 2x 1 3y 5 5

6(4) 1 5(21) 0 19 2(4) 1 3(21) 0 5

19 5 19 ✓ 5 5 5 ✓

E X A M P L E 1 Multiply one equation, then add

ANOTHER WAY

You can also multiplyEquation 2 by 3 andsubtract the equations.

3 (23)

Before You solved linear systems by adding or subtracting.

Now You will solve linear systems by multiplying first.

Why So you can solve a problem about preparing food, as in Ex. 39.


E X A M P L E 2 Multiply both equations, then subtract

Solve the linear system: 4x 1 5y 5 35 Equation 1 2y 5 3x 2 9 Equation 2

Solution

STEP 1 Arrange the equations so that like terms are in columns.


23x 1 2y 5 29 Rewrite Equation 2.

STEP 2 Multiply Equation 1 by 2 and Equation 2 by 5 so that the coeffcientof y in each equation is the least common multiple of 5 and 2, or 10.

4x 1 5y 5 35 8x 1 10y 5 70

23x 1 2y 5 29 215x 1 10y 5 245

STEP 3 Subtract the equations. 23x 5 115

STEP 4 Solve for x. x 5 5

STEP 5 Substitute 5 for x in either of the original equations and solve for y.


4(5) 1 5y 5 35 Substitute 5 for x.

y 5 3 Solve for y.




4x 1 5y 5 35 2y 5 3x 2 9

4(5) 1 5(3) 0 35 2(3) 0 3(5) 2 9

35 5 35 ✓ 6 5 6 ✓

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MULTIPLYING BOTH EQUATIONS To eliminate one variable when adding orsubtracting equations in a linear system, you may need to multiply bothequations by constants. Use the least common multiple of the coefficients ofone of the variables to determine the constants.

2x 2 9y 5 1 8x 2 36y 5 4

7x 2 12y 5 23 21x 2 36y 5 693 3

3 4

3 2

3 5


Solve the linear system using elimination.

1. 6x 2 2y 5 1 2. 2x 1 5y 5 3 3. 3x 2 7y 5 522x 1 3y 5 25 3x 1 10y 5 23 9y 5 5x 1 5

ANOTHER WAY

You can also multiplyEquation 1 by 3 andEquation 2 by 4.Then add the revisedequations to eliminate x.

The least common multiple of 29 and 212 is 236.



4. SOCCER A sports equipment store is having a sale on soccer balls. Asoccer coach purchases 10 soccer balls and 2 soccer ball bags for $155.Another soccer coach purchases 12 soccer balls and 3 soccer ball bagsfor $189. Find the cost of a soccer ball and the cost of a soccer ball bag.

Darlene is making a quilt that has alternating stripes of regular quiltingfabric and sateen fabric. She spends $76 on a total of 16 yards of thetwo fabrics at a fabric store. Which system of equations can be used tofind the amount x (in yards) of regular quilting fabric and the amounty (in yards) of sateen fabric she purchased?

A x 1 y 5 16 B x 1 y 5 16x 1 y 5 76 4x 1 6y 5 76

C x 1 y 5 76 D x 1 y 5 164x 1 6y 5 16 6x 1 4y 5 76

★

Solution

Write a system of equations where x is the number of yards of regular quiltingfabric purchased and y is the number of yards of sateen fabric purchased.

Equation 1: Amount of fabric

Amountof quilting

fabric1

Amountof sateen

fabric5

Totalyards offabric

x 1 y 5 16

Equation 2: Cost of fabric

Quiltingfabric price(dollars/yd)

pAmount

of quiltingfabric (yd)

1Sateen

fabric price(dollars/yd)

pAmountof sateenfabric (yd)

5Totalcost

(dollars)

4 p x 1 6 p y 5 76

The system of equations is: x 1 y 5 16 Equation 14x 1 6y 5 76 Equation 2

c The correct answer is B. A B C D

E X A M P L E 3 Standardized Test Practice

ELIMINATE CHOICES

You can eliminatechoice A because x + ycannot equal both 16and 76.

Sateen fabric costs $6 per yard.

Quilting fabric costs $4 per yard.


1. VOCABULARY What is the least common multiple of 12 and 18?

2. ★ WRITING Explain how to solve the linearsystem using the elimination method.


3. x 1 y 5 2 4. 3x 2 2y 5 3 5. 4x 1 3y 5 82x 1 7y 5 9 2x 1 y 5 1 x 2 2y 5 13

6. 10x 2 9y 5 46 7. 8x 2 5y 5 11 8. 11x 2 20y 5 2822x 1 3y 5 10 4x 2 3y 5 5 3x 1 4y 5 36

7.4 EXERCISES

CONCEPT SUMMARY For Your Notebook

Methods for Solving Linear Systems

Method

Table (p. 426)

Graphing (p. 427)

Substitution (p. 435)

Addition (p. 444)

Subtraction (p. 445)

Multiplication (p. 451)

Example

y 5 4 2 2x4x 1 2y 5 8

4x 1 7y = 156x 2 7y = 5

3x 1 5y = 2133x 1 y = 25

9x 1 2y = 383x 2 5y = 7

When to Use

When x-values are integers, so thatequal values can be seen in the table

When you want to see the lines that theequations represent

When one equation is already solvedfor x or y

When the coefficients of one variableare opposites

When the coefficients of one variableare the same

When no corresponding coefficients arethe same or opposites

EXAMPLE 1




x y 5 2x y 5 3x 2 1

0 0 21

1 2 2

2 4 5

x

y

3x 2 2y 5 2

x 1 y 5 41

1

HOMEWORKKEY




SKILL PRACTICE



9. 4x 2 3y 5 8 10. 22x 2 5y 5 9 11. 7x 2 6y 5 215x 2 2y 5 211 3x 1 11y 5 4 5x 2 4y 5 1

12. 7x 1 3y 5 212 13. 9x 2 8y 5 4 14. 12x 2 7y 5 222x 1 5y 5 38 2x 2 3y 5 24 28x 1 11y 5 14

15. 9x 1 2y 5 39 16. 27x 1 10y 5 11 17. 214x 1 15y 5 156x 1 13y 5 29 28x 1 15y 5 34 21x 2 20y 5 210

18. ★ MULTIPLE CHOICE Which ordered pair is a solution of the linear system15x 1 8y 5 6 and 25x 1 12y 5 14?

A (23, 22) B (23, 2) C (22, 23) D (2, 23)

ERROR ANALYSIS Describe and correct the error when solving the linearsystem.

19.2x 2 3y 5 29 4x 2 6y 5 2185x 2 6y 5 29 5x 2 6y 5 29

9x 5227

x 5 23

20.9x 1 8y 5 11 27x 1 24y 5 117x 1 6y 5 9 28x 1 24y 5 9

2x 5 2

x 5 22

SOLVING LINEAR SYSTEMS Solve the linear system using any algebraicmethod.

21. 3x 1 2y 5 4 22. 4x 2 5y 5 18 23. 8x 2 9y 5 2152y 5 8 2 5x 3x 5 y 1 11 24x 5 19 1 y

24. 0.3x 1 0.1y 5 20.1 25. 4.4x 2 3.6y 5 7.6 26. 3x 2 2y 5 2202x 1 y 5 3 x 2 y 5 1 x 1 1.2y 5 6.4

27. 0.2x 2 1.5y 5 21 28. 1.5x 2 3.5y 5 25 29. 4.9x 1 2.4y 5 7.4x 2 4.5y 5 1 21.2x 1 2.5y 5 1 0.7x 1 3.6y 5 22.2

30. x 1 y 5 0 31. 3x 1 y 5 1}3

32. 3}5

x 2 3}4

y 5 23

1}2

x 2 1}2

y 5 2 2x 2 3y 5 8}3

2}5

x 1 1}3

y 5 8

33. GEOMETRY A rectangle has a perimeter of 18 inches.A new rectangle is formed by doubling the width w andtripling the length l, as shown. The new rectangle has aperimeter P of 46 inches.

a. Write and solve a system of linear equations to findthe length and width of the original rectangle.

b. Find the length and width of the new rectangle.

34. ★ WRITING For which values of a can you solve the linear systemax 1 3y 5 2 and 4x 1 5y 5 6 without multiplying first? Explain.

CHALLENGE Find the values of a and b so that thelinear system has the given solution.

35. (4, 2) 36. (2, 1)

EXAMPLE 2

on p. 452 forExs. 9–20

32 33

34

3l

2wP 5 46 in.


bx 2 ay 5 10 Equation 2


EXAMPLE 3

on p. 453for Exs. 37–39

PROBLEM SOLVING


5 MULTIPLEREPRESENTATIONS


37. BOOK SALE A library is having a book sale to raise money. Hardcoverbooks cost $4 each and paperback books cost $2 each. A person spends$26 for 8 books. How many hardcover books did she purchase?

38. MUSIC A website allows users to download individual songs or an entirealbum. All individual songs cost the same to download, and all albums costthe same to download. Ryan pays $14.94 to download 5 individual songs and1 album. Seth pays $22.95 to download 3 individual songs and 2 albums.How much does the website charge to download a song? an entire album?

39. FARM PRODUCTS The table shows the number of apples needed to makethe apple pies and applesauce sold at a farm store. During a recent applepicking at the farm, 169 Granny Smith apples and 95 Golden Deliciousapples were picked. How many apple pies and batches of applesauce canbe made if every apple is used?

Type of apple Granny Smith Golden Delicious

Needed for a pie 5 3

Needed for a batch of applesauce 4 2

40. MULTIPLE REPRESENTATIONS Tickets for admission to a high schoolfootball game cost $3 for students and $5 for adults. During one game,$2995 was collected from the sale of 729 tickets.

a. Writing a System Write and solve a system of linear equations to findthe number of tickets sold to students and the number of tickets soldto adults.

b. Drawing a Graph Graph the system of linear equations. Use thegraph to determine whether your answer to part (a) is reasonable.

41. ★ SHORT RESPONSE A dim sumrestaurant offers two sizes of dishes:small and large. All small dishescost the same and all large dishescost the same. The bills show thecost of the food before the tip isincluded. What will 3 small and2 large dishes cost before the tipis included? Explain.

42. ★ OPEN – ENDED Describe areal-world problem that can be solvedusing a system of linear equations. Thensolve the problem and explain what thesolution means in this situation.

457

43. INVESTMENTS Matt invested $2000 in stocks and bonds. This yearthe bonds paid 8% interest, and the stocks paid 6% in dividends. Mattreceived a total of $144 in interest and dividends. How much money didhe invest in stocks? in bonds?

44. CHALLENGE You drive a car 45 miles at an average speed r (in miles perhour) to reach your destination. Due to traffic, your average speed on the

return trip is 3}4

r. The round trip took a total of 1 hour 45 minutes. Find the

average speed for each leg of your trip.

Solve the linear system using elimination. (pp. 444, 451)

1. x 1 y 5 4 2. 2x 2 y 5 2 3. x 1 y 5 523x 1 y 5 28 6x 2 y 5 22 2x 1 y 5 23

4. x 1 3y 5 210 5. x 1 3y 5 10 6. x 1 7y 5 102x 1 5y 5 230 3x 2 y 5 13 x 1 2y 5 28

7. 4x 2 y 5 22 8. x 1 3y 5 1 9. 3x 1 y 5 213x 1 2y 5 7 5x 1 6y 5 14 x 1 y 5 1

10. 2x 2 3y 5 25 11. 7x 1 2y 5 13 12. 1}3

x 1 5y 5 235x 1 2y 5 16 4x 1 3y 5 13

22}3

x 1 6y 5 210



Graph the equation. (pp. 215, 225, 244)

45. x 2 8y 5 210 46. 22x 1 5y 5 215 47. 1}4

x 1 1}2

y 5 8

48. y 5 2}3

x 2 1 49. y 5 29x 1 2 50. y 2 1 5 2x 2 7

Determine which lines are parallel. (p. 244)

51.

x

y

2

2

(3, 3)

(2, 5)(0, 3)

(0, 1)

(6, 3)

(3, 1)

bbaa cc 52.

x

ybb aa2

1

(5, 22.5)

(2, 2)

(6, 0)(0, 1)

(0, 22)(22, 21)

cc

Solve the linear system using any method. (pp. 427, 435, 444, 451)

53. y 5 4x 2 9 54. y 5 22x 1 20 55. x 1 2y 5 0y 5 28x 1 15 y 5 26x 1 40 2x 5 y 1 3

56. x 1 2y 5 2 57. 7x 2 8y 5 215 58. 8x 1 y 5 52x 1 y 5 211 5x 1 8y 5 3 22x 1 y 5 0

MIXED REVIEW

PREVIEW



Lessons 7.1–7.4

MIXED REVIEW of Problem SolvingMIXED REVIEW of Problem Solving STATE TEST PRACTICEclasszone.com

1. MULTI-STEP PROBLEM Flying into the wind,a helicopter takes 15 minutes to travel15 kilometers. The return flight takes12 minutes. The wind speed remainsconstant during the trip.

a. Find the helicopter’s average speed (inkilometers per hour) for each leg of the trip.

b. Write a system of linear equations thatrepresents the situation.

c. What is the helicopter’s average speed instill air? What is the speed of the wind?

2. SHORT RESPONSE At a grocery store, acustomer pays a total of $9.70 for 1.8 poundsof potato salad and 1.4 pounds of coleslaw.Another customer pays a total of $6.55 for1 pound of potato salad and 1.2 pounds ofcoleslaw. How much do 2 pounds of potatosalad and 2 pounds of coleslaw cost? Explain.

3. GRIDDED ANSWER During one day, twocomputers are sold at a computer store. Thetwo customers each arrange payment planswith the salesperson. The graph shows theamount y of money (in dollars) paid for thecomputers after x months. After how manymonths will each customer have paid thesame amount?

Months since purchase

Amou

nt p

aid

(dol

lars

)

2 4 6 8 10

400

200

00 x

y

4. OPEN-ENDED Describe a real-world problemthat can be modeled by a linear system.Then solve the system and interpret thesolution in the context of the problem.

5. SHORT RESPONSE A hot air balloon islaunched at Kirby Park, and it ascends ata rate of 7200 feet per hour. At the sametime, a second hot air balloon is launchedat Newman Park, and it ascends at a rate of4000 feet per hour. Both of the balloons stopascending after 30 minutes. The diagramshows the altitude of each park. Are the hotair balloons ever at the same height at thesame time? Explain.

6. EXTENDED RESPONSE A chemist needs500 milliliters of a 20% acid and 80% watermix for a chemistry experiment. The chemistcombines x milliliters of a 10% acid and90% water mix and y milliliters of a 30% acidand 70% water mix to make the 20% acid and80% water mix.

a. Write a linear system that represents thesituation.

b. How many milliliters of the 10% acidand 90% water mix and the 30% acid and70% water mix are combined to make the20% acid and 80% water mix?

c. The chemist also needs 500 milliliters ofa 15% acid and 85% water mix. Does thechemist need more of the 10% acid and90% water mix than the 30% acid and70% water mix to make this new mix?Explain.

7.5 Solve Special Types of Linear Systems 459

7.5 Solve Special Types ofLinear Systems

Key Vocabulary• inconsistent system• consistent

dependent system• system of linear

equations, p. 427

• parallel, p. 244

A linear system can have no solution or infinitely many solutions. A linearsystem has no solution when the graphs of the equations are parallel.A linear system with no solution is called an inconsistent system.

A linear system has infinitely many solutions when the graphs of theequations are the same line. A linear system with infinitely many solutionsis called a consistent dependent system.

Show that the linear system has no solution.

3x 1 2y 5 10 Equation 1 3x 1 2y 5 2 Equation 2

Solution

METHOD 1 Graphing

Graph the linear system.

x

y

3x 1 2y 5 2

3x 1 2y 5 10

2

22

23

23

21

c The lines are parallel because they have the same slope but differenty-intercepts. Parallel lines do not intersect, so the system has no solution.

METHOD 2 Elimination

Subtract the equations.

3x 1 2y 5 10 3x 1 2y 5 2

0 5 8 This is a false statement.

c The variables are eliminated and you are left with a false statementregardless of the values of x and y. This tells you that the system hasno solution.

at classzone.com

E X A M P L E 1 A linear system with no solution

Before You found the solution of a linear system.

Now You will identify the number of solutions of a linear system.

Why? So you can compare distances traveled, as in Ex. 39.

REVIEW GRAPHING

For help with graphinglinear equations, seepp. 215, 225, and 244.

IDENTIFY TYPES OFSYSTEMS

The linear system inExample 1 is called aninconsistent systembecause the lines donot intersect (are notconsistent).


Show that the linear system has infinitely many solutions.


y 5 1}2

x 1 2 Equation 2

Solution

METHOD 1 Graphing


x

y

y 5 x 1 212

x 2 2y 5 24

1

1

c The equations represent the same line, so any point on the line is asolution. So, the linear system has infinitely many solutions.

METHOD 2 Substitution

Substitute 1}2

x 1 2 for y in Equation 1 and solve for x.


x 2 211}2

x 1 22 5 24 Substitute 1}2

x 1 2 for y.

24 5 24 Simplify.

c The variables are eliminated and you are left with a statement that istrue regardless of the values of x and y. This tells you that the system hasinfinitely many solutions.

E X A M P L E 2 A linear system with infinitely many solutions


Tell whether the linear system has no solution or infinitely many solutions.Explain.

1. 5x 1 3y 5 6 2. y 5 2x 2 425x 2 3y 5 3 26x 1 3y 5 212

IDENTIFYING THE NUMBER OF SOLUTIONS When the equations of a linearsystem are written in slope-intercept form, you can identify the number ofsolutions of the system by looking at the slopes and y-intercepts of the lines.

Number of solutions Slopes and y-intercepts

One solution Different slopes

No solutionSame slope

Different y-intercepts

Infinitely many solutionsSame slope

Same y-intercept

IDENTIFY TYPES OFSYSTEMS

The linear system inExample 2 is called aconsistent dependentsystem because thelines intersect (areconsistent) and theequations are equivalent(are dependent).


E X A M P L E 3 Identify the number of solutions

Without solving the linear system, tell whether the linear system has onesolution, no solution, or infinitely many solutions.

a. 5x 1 y 5 22 Equation 1 b. 6x 1 2y 5 3 Equation 1 –10x 2 2y 5 4 Equation 2 6x 1 2y 5 25 Equation 2

Solution

a. y 5 25x 2 2 Write Equation 1 in slope-intercept form.

y 5 25x 2 2 Write Equation 2 in slope-intercept form.

c Because the lines have the same slope and the same y-intercept,the system has infinitely many solutions.

b. y 5 23x 1 3}2

Write Equation 1 in slope-intercept form.

y 5 23x 2 5}2

Write Equation 2 in slope-intercept form.

c Because the lines have the same slope but different y-intercepts,the system has no solution.


3. Without solving the linear system, tellwhether it has one solution, no solution,or infinitely many solutions.

4. WHAT IF? In Example 4, suppose a glossy print costs $3 more than aregular print. Find the cost of a glossy print.

E X A M P L E 4 Write and solve a system of linear equations

ART An artist wants to sell prints of herpaintings. She orders a set of prints for each oftwo of her paintings. Each set contains regularprints and glossy prints, as shown in the table.Find the cost of one glossy print.

Solution

STEP 1 Write a linear system. Let x be the cost (in dollars) of a regular print,and let y be the cost (in dollars) of a glossy print.

45x 1 30y 5 465 Cost of prints for one painting 15x 1 10y 5 155 Cost of prints for other painting

STEP 2 Solve the linear system using elimination.

45x 1 30y 5 465 45x 1 30y 5 465 15x 1 10y 5 155 245x 2 30y 5 2465

0 5 0

c There are infinitely many solutions, so you cannot determine the cost ofone glossy print. You need more information.

Regular Glossy Cost

45 30 $465

15 10 $155

3 (23)

x 2 3y 5 215 Equation 12x 2 3y 5 218 Equation 2


1. VOCABULARY Copy and complete: A linear system with no solution iscalled a(n) ? system.

2. VOCABULARY Copy and complete: A linear system with infinitely manysolutions is called a(n) ? system.

3. ★ WRITING Describe the graph of a linear system that has no solution.

4. ★ WRITING Describe the graph of a linear system that has infinitelymany solutions.

INTERPRETING GRAPHS Match the linear system with its graph. Then usethe graph to tell whether the linear system has one solution, no solution, orinfinitely many solutions.

5. x 2 3y 5 29 6. x 2 y 5 24 7. x 1 3y 5 21x 2 y 5 21 23x 1 3y 5 2 22x 2 6y 5 2

A.

x

y

1

2

B.

x

y

2

1

C.

x

y

2

1

7.5 EXERCISES

CONCEPT SUMMARY For Your Notebook

Number of Solutions of a Linear System

One solution

x

y

The lines intersect.

The lines havedifferent slopes.

No solution

x

y

The lines are parallel.

The lines have the sameslope and differenty-intercepts.

Infinitely many solutions

x

y

The lines coincide.

The lines have the sameslope and the samey-intercept.

HOMEWORKKEY


★ 5 STANDARDIZED TEST PRACTICEExs. 3, 4, 24, 25, 32, 33, and 40

SKILL PRACTICE


INTERPRETING GRAPHS Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions.

8. x 1 y 5 22 9. 3x 2 4y 5 12 10. 3x 2 y 5 29y 5 2x 1 5

y 5 3}4

x 2 3 3x 1 5y 5 215

11. 22x 1 2y 5 216 12. 29x 1 6y 5 18 13. 23x 1 4y 5 123x 2 6y 5 30 6x 2 4y 5 212 23x 1 4y 5 24

14. ERROR ANALYSIS Describe andcorrect the error in solving the linear system below.

6x 1 y 5 365x 2 y 5 8

SOLVING LINEAR SYSTEMS Solve the linear system using substitution or elimination.

15. 2x 1 5y 5 14 16. 216x 1 2y 5 22 17. 3x 2 2y 5 256x 1 7y 5 10 y 5 8x 2 1 4x 1 5y 5 47

18. 5x 2 5y 5 23 19. x 2 y 5 0 20. x 2 2y 5 7y 5 x 1 0.6 5x 2 2y 5 6 2x 1 2y 5 7

21. 218x 1 6y 5 24 22. 4y 1 5x 5 15 23. 6x 1 3y 5 93x 2 y 5 22 x 5 8y 1 3 2x 1 9y 5 27

24. ★ MULTIPLE CHOICE Which of the linear systems has exactly one solution?

A 2x 1 y 5 9 B 2x 1 y 5 9x 2 y 5 9 x 2 y 5 29

C 2x 1 y 5 9 D x 2 y 5 292x 2 y 5 9 2x 1 y 5 29

25. ★ MULTIPLE CHOICE Which of the linear systems has infinitely many solutions?

A 15x 1 5y 5 20 B 15x 2 5y 5 20 6x 2 2y 5 8 6x 2 2y 5 28

C 15x 2 5y 5 220 D 15x 2 5y 5 20 6x 2 2y 5 8 6x 2 2y 5 8

IDENTIFYING THE NUMBER OF SOLUTIONS Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions.

26. y 5 26x 2 2 27. y 5 7x 1 13 28. 4x 1 3y 5 2712x 1 2y 5 26 221x 1 3y 5 39 4x 2 3y 5 227

29. 9x 2 15y 5 24 30. 0.3x 1 0.4y 5 2.4 31. 0.9x 2 2.1y 5 12.36x 2 10y 5 16 0.5x 2 0.6y 5 0.2 1.5x 2 3.5y 5 20.5

EXAMPLES 1 and 2

on pp. 459–460 for Exs. 8–25

EXAMPLE 3

on p. 461 for Exs. 26–31

x

y

1

2

The lines do not intersect, so there is no solution.

464 ★ 5 STANDARDIZED

TEST PRACTICE 5 WORKED-OUT SOLUTIONS

on p. WS1

EXAMPLE 4

on p. 461for Exs. 36–38

32. ★ OPEN – ENDED Write a linear system so that it has infinitely manysolutions, and one of the equations is y 5 3x 1 2.

33. ★ OPEN – ENDED Write a linear system so that it has no solution andone of the equations is 7x 2 8y 5 29.

34. REASONING Give a counterexample for the following statement:If the graphs of the equations of a linear system have the same slope,then the linear system has no solution.

35. CHALLENGE Find values of p, q, and r that produce the solution(s).

a. No solution

b. Infinitely many solutions

c. One solution of (4, 1)

PROBLEM SOLVING

36. RECREATION One admission to a roller skating rink costs x dollars andrenting a pair of skates costs y dollars. A group pays $243 for admissionfor 36 people and 21 skate rentals. Another group pays $81 for admissionfor 12 people and 7 skate rentals. Is there enough information todetermine the cost of one admission to the roller skating rink? Explain.

37. TRANSPORTATION A passenger train travels from New York City toWashington, D.C., then back to New York City. The table shows thenumber of coach tickets and business class tickets purchased for eachleg of the trip. Is there enough information to determine the cost of onecoach ticket? Explain.

38. PHOTOGRAPHY In addition to taking pictures on your digital camera,you can record 30 second movies. All pictures use the same amount ofmemory, and all 30 second movies use the same amount of memory. Thenumber of pictures and 30 second movies on 2 memory cards is shown.

a. Is there enough information givento determine the amount of memoryused by a 30 second movie? Explain.

b. Given that a 30 second movie uses50 times the amount of memorythat a digital picture uses, can youdetermine the amount of memoryused by a 30 second movie? Explain.

px 1 qy 5 r Equation 12x 2 3y 5 5 Equation 2

Destination Coachtickets

Business classtickets

Money collected(dollars)

Washington, D.C. 150 80 22,860

New York City 170 100 27,280

Size of card(megabytes) 64 256

Pictures 450 1800

Movies 7 28

465

39. MULTI-STEP PROBLEM Two people aretraining for a speed ice-climbing event.During a practice climb, one climber starts15 seconds after the first climber. The ratesthat the climbers ascend are shown.

a. Let d be the distance (in feet) traveled bya climber t seconds after the first personstarts climbing. Write a linear system thatmodels the situation.

b. Graph the linear system from part (a).Does the second climber catch up to thefirst climber? Explain.

40. ★ EXTENDED RESPONSE Two employees at a banquet facility are giventhe task of folding napkins. One person starts folding napkins at a rate of5 napkins per minute. The second person starts 10 minutes after the firstperson and folds napkins at a rate of 4 napkins per minute.

a. Model Let y be the number of napkins folded x minutes after the firstperson starts folding. Write a linear system that models the situation.

b. Solve Solve the linear system.

c. Interpret Does the solution of the linear system make sense in thecontext of the problem? Explain.

41. CHALLENGE An airplane has an average air speed of 160 miles perhour. The airplane takes 3 hours to travel with the wind from Salem toLancaster. The airplane has to travel against the wind on the return trip.After 3 hours into the return trip, the airplane is 120 miles from Salem.Find the distance from Salem to Lancaster. If the problem cannot besolved with the information given, explain why.


Climbs 10 feetevery 30 seconds

Climbs 5 feetevery 15 seconds

Solve the equation, if possible. (p. 154)

42. 61 1 5c 5 7 2 4c 43. 3m 2 2 5 7m 2 50 1 8m

44. 11z 1 3 5 10(2z 1 3) 45. 26(1 2 w) 5 14(w 2 5)

Solve the inequality. Then graph your solution.

46. x 1 15 < 23 (p. 356) 47. x 2 1 ≥ 10 (p. 356)

48. x}4

> 22.5 (p. 363) 49. 27x ≤ 84 (p. 363)

50. 2 2 5x ≥ 27 (p. 369) 51. 3x 2 9 > 3(x 2 3) (p. 369)

52. 22 < x 1 3 ≤ 11 (p. 380) 53. 27 ≤ 5 2 2x ≤ 7 (p. 380)

54. ⏐x 2 3⏐≥ 5 (p. 398) 55. 2⏐2x 2 1⏐2 9 ≤ 1 (p. 398)

Graph the inequality. (p. 405)

56. x 1 y < 23 57. x 2 y ≥ 1 58. 2x 2 y < 5

59. 22x 2 3y ≤ 9 60. y ≤ 24 61. x > 6.5

PREVIEW


MIXED REVIEW



inequalities• solution of a

system of linearinequalities

• graph of a systemof linearinequalities

7.6 Solve Systems ofLinear Inequalities

Graph the system of inequalities. y > 2x 2 2 Inequality 1y ≤ 3x 1 6 Inequality 2

Solution

Graph both inequalities in the same coordinateplane. The graph of the system is the intersectionof the two half-planes, which is shown as thedarker shade of blue.

CHECK Choose a point in the dark blue region,such as (0, 1). To check this solution, substitute0 for x and 1 for y into each inequality.

1 >? 0 2 2 1 ≤? 0 1 6

1 > 22 ✓ 1 ≤ 6 ✓

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E X A M P L E 1 Graph a system of two linear inequalities

A system of linear inequalities in two variables, or simply a system ofinequalities, consists of two or more linear inequalities in the samevariables. An example is shown.

x 2 y > 7 Inequality 1 2x 1 y < 8 Inequality 2

A solution of a system of linear inequalities is an ordered pair that is asolution of each inequality in the system. For example, (6, 25) is a solutionof the system above. The graph of a system of linear inequalities is thegraph of all solutions of the system.


Graphing a System of Linear Inequalities

STEP 1 Graph each inequality (as you learned to do in Lesson 6.7).

STEP 2 Find the intersection of the half-planes. The graph of the systemis this intersection.

x

y

1

1

(0, 1)

REVIEW GRAPHINGINEQUALITIES

For help with graphinga linear inequality intwo variables, seep. 405.

Before You graphed linear inequalities in two variables.

Now You will solve systems of linear inequalities in two variables.

Why So you can find a marching band’s competition score, as in Ex. 36.

7.6 Solve Systems of Linear Inequalities 467

E X A M P L E 2 Graph a system of three linear inequalities

Graph the system of inequalities. y ≥ 21 Inequality 1x > 22 Inequality 2x 1 2y ≤ 4 Inequality 3

Solution

Graph all three inequalities in the same coordinate plane. The graph of thesystem is the triangular region shown.

THE SOLUTION REGION In Example 1, the half-plane for each inequality isshaded, and the solution region is the intersection of the half-planes. Fromthis point on, only the solution region will be shaded.


Graph the system of linear inequalities.

1. y < x 2 4 2. y > 2x 1 2 3. y > 2xy > 2x 1 3 y < 4 y > x 2 4

x < 3 y < 5

Write a system of inequalities for the shaded region.

Solution

INEQUALITY 1: One boundary line for the shadedregion is y 5 3. Because the shaded region isabove the solid line, the inequality is y ≥ 3.

INEQUALITY 2: Another boundary line for theshaded region has a slope of 2 and a y-interceptof 1. So, its equation is y 5 2x 1 1. Because theshaded region is above the dashed line, theinequality is y > 2x 1 1.

c The system of inequalities for the shaded region is:

y ≥ 3 Inequality 1 y > 2x 1 1 Inequality 2

E X A M P L E 3 Write a system of linear inequalities

x

y

2

1

REVIEW EQUATIONSOF LINES

For help with writing anequation of a line, seepp. 283, 302, and 311.

x

y

1

1

The region is tothe right of theline x 5 22.

The region is onand below theline x 1 2y 5 4.

The region is onand above theline y 5 21.


E X A M P L E 4 Write and solve a system of linear inequalities

BASEBALL The National Collegiate Athletic Association (NCAA)regulates the lengths of aluminum baseball bats used by collegebaseball teams. The NCAA states that the length (in inches) ofthe bat minus the weight (in ounces) of the bat cannot exceed 3.Bats can be purchased at lengths from 26 to 34 inches.

a. Write and graph a system of linear inequalities thatdescribes the information given above.

b. A sporting goods store sells an aluminum bat that is 31 incheslong and weighs 25 ounces. Use the graph to determine ifthis bat can be used by a player on an NCAA team.

Solution

a. Let x be the length (in inches) of the bat, and let y be the weight(in ounces) of the bat. From the given information, you can writethe following inequalities:

x 2 y ≤ 3 The difference of the bat’s length and weight can be at most 3.

x ≥ 26 The length of the bat must be at least 26 inches.

x ≤ 34 The length of the bat can be at most 34 inches.

y ≥ 0 The weight of the bat cannot be a negative number.

Graph each inequality in the system. Thenidentify the region that is common to all ofthe graphs of the inequalities. This regionis shaded in the graph shown.

b. Graph the point that represents a bat thatis 31 inches long and weighs 25 ounces.

c Because the point falls outside thesolution region, the bat cannot be usedby a player on an NCAA team.


Write a system of inequalities that defines the shaded region.

4. 5.

6. WHAT IF? In Example 4, suppose a Senior League (ages 10–14) playerwants to buy the bat described in part (b). In Senior League, the length(in inches) of the bat minus the weight (in ounces) of the bat cannotexceed 8. Write and graph a system of inequalities to determine whetherthe described bat can be used by the Senior League player.

x

y

(31, 25)

5

10

WRITING SYSTEMSOF INEQUALITIES

Consider the valuesof the variables whenwriting a system ofinequalities. In manyreal-world problems,the values cannot benegative.

x

y

1

2

x

y

1

1

x


EXAMPLE 2

on p. 467for Exs. 18221

EXAMPLE 1

on p. 466for Exs. 6217

1. VOCABULARY Copy and complete: A(n) ? of a system of linearinequalities is an ordered pair that is a solution of each inequality inthe system.

2. ★ WRITING Describe the steps you would take tograph the system of inequalities shown.

CHECKING A SOLUTION Tell whether the ordered pair is a solution of thesystem of inequalities.

3. (1, 1) 4. (0, 6) 5. (3, 21)

x

y

1

1

x

y

1

1

x

y

1

1

MATCHING SYSTEMS AND GRAPHS Match the system of inequalities withits graph.

6. x 2 4y > 28 7. x 2 4y ≥ 28 8. x 2 4y > 28x ≥ 2 x < 2 y ≥ 2

A.

x

y

1

1

B.

x

y

1

1

C.

x

y

1

1

GRAPHING A SYSTEM Graph the system of inequalities.

9. x > 25 10. y ≤ 10 11. x > 3x < 2 y ≥ 6 y > x

12. y < 22x 1 3 13. y ≥ 0 14. y ≥ 2x 1 1y ≥ 4 y < 2.5x 2 1 y < 2x 1 4

15. x < 8 16. y ≥ 22 17. y 2 2x < 7x 2 4y ≤ 28 2x 1 3y > 26 y 1 2x > 21

18. x < 4 19. x ≥ 0 20. x 1 y ≤ 10y > 1 y ≥ 0 x 2 y ≥ 2y ≥ 2x 1 1 6x 2 y < 12 y ≥ 2

21. ★ MULTIPLE CHOICE Which ordered pair is a solution of the system2x 2 y ≤ 5 and x 1 2y > 2?

A (1, 21) B (4, 1) C (2, 0) D (3, 2)

7.6 EXERCISES

x 2 y < 7 Inequality 1y ≥ 3 Inequality 2

HOMEWORKKEY



SKILL PRACTICE

470

22. ★ MULTIPLE CHOICE The graph of whichsystem of inequalities is shown?

A y < 2x B y < 2x 2x 1 3y < 6 2x 1 3y > 6

C y > 2x D y > 2x 2x 1 3y < 6 2x 1 3y > 6

23. ERROR ANALYSIS Describe and correctthe error in graphing this system ofinequalities:

x 1 y < 3 Inequality 1x > 21 Inequality 2x ≤ 3 Inequality 3

WRITING A SYSTEM Write a system of inequalities for the shaded region.

24.

x

y

1

2

25.

x

y

1

1

26.

x

y

1

1

27.

x

y

1

3

28.

x

y

3

1

29.

x

y

211

GRAPHING A SYSTEM Graph the system of inequalities.

30. x > 4 31. x 1 y < 4 32. x ≤ 10x < 9 x 1 y > 22 3x 1 2y ≥ 9y ≤ 2 x 2 y ≤ 3 x 2 2y ≤ 6y > 22 x 2 y ≥ 24 x 1 y ≤ 5

33. ★ SHORT RESPONSE Does the system of inequalities have any solutions?Explain.

x 2 y > 5 Inequality 1x 2 y < 1 Inequality 2

CHALLENGE Write a system of inequalities for the shaded region described.

34. The shaded region is a rectangle with vertices at (2, 1), (2, 4), (6, 4), and(6, 1).

35. The shaded region is a triangle with vertices at (23, 0), (3, 2), and (0, 22).

x

y

1

1

x

y

1

1



EXAMPLE 3

on p. 467for Exs. 24–29

EXAMPLE 2

on p. 467for Exs. 22–23


PROBLEM SOLVING

36. COMPETITION SCORES In a marching band competition, scoring is basedon a musical evaluation and a visual evaluation. The musical evaluationscore cannot exceed 60 points, the visual evaluation score cannot exceed40 points. Write and graph a system of inequalities for the scores that amarching band can receive.

37. NUTRITION For a hiking trip, you are making a mix of x ounces ofpeanuts and y ounces of chocolate pieces. You want the mix to have lessthan 70 grams of fat and weigh less than 8 ounces. An ounce of peanutshas 14 grams of fat, and an ounce of chocolate pieces has 7 grams of fat.Write and graph a system of inequalities that models the situation.

38. FISHING LIMITS You are fishing in a marina for surfperch and rockfish,which are two species of bottomfish. Gaming laws in the marina allowyou to catch no more than 15 surfperch per day, no more than 10 rockfishper day, and no more than 15 total bottomfish per day.

a. Write and graph a systemof inequalitites that modelsthe situation.

b. Use the graph to determinewhether you can catch11 surfperch and 9 rockfishin one day.

39. HEALTH A person’s maximum heart rate (in beats per minute) is givenby 220 2 x where x is the person’s age in years (20 ≤ x ≤ 65). Whenexercising, a person should aim for a heart rate that is at least 70% ofthe maximum heart rate and at most 85% of the maximum heart rate.

a. Write and graph a system of inequalitites that models the situation.

b. A 40-year-old person’s heart rate varies from 104 to 120 beats perminute while exercising. Does his heart rate stay in the suggestedtarget range for his age? Explain.

40. ★ SHORT RESPONSE A photography shop has a self-service photo centerthat allows you to make prints of pictures. Each sheet of printed picturescosts $8. The number of pictures that fit on each sheet is shown.

a. You want at least 16 picturesof any size, and you are willingto spend up to $48. Write andgraph a system of inequalitiesthat models the situation.

b. Will you be able to purchase12 pictures that are 3 inchesby 5 inches and 6 picturesthat are 4 inches by 6 inches?Explain.

EXAMPLE 4

on p. 468for Exs. 36–38

Two 4 inch by 6 inchpictures fit on one sheet.

Four 3 inch by 5 inchpictures fit on one sheet.

Surfperch Rockfish

472


Graph the linear system. Then use the graph to tell whether the linearsystem has one solution, no solution, or infinitely many solutions. (p. 459)

1. x 2 y 5 1 2. 6x 1 2y 5 16 3. 3x 2 3y 5 22x 2 y 5 6 2x 2 y 5 2 26x 1 6y 5 4

Graph the system of linear inequalities. (p. 466)

4. x > 23 5. y ≤ 2 6. 4x ≥ yx < 7 y < 6x 1 2 2x 1 4y < 4

7. x 1 y < 2 8. y ≥ 3x 2 4 9. x > 252x 1 y > 23 y ≤ x x < 0y ≥ 0 y ≥ 25x 2 15 y ≤ 2x 1 7


41. CHALLENGE You make necklaces and keychains to sell at a craft fair. Thetable shows the time that it takes to make each necklace and keychain,the cost of materials for each necklace and keychain, and the time andmoney that you can devote to making necklaces and keychains.

Necklace Keychain Available

Time to make (hours) 0.5 0.25 20

Cost to make (dollars) 2 3 120

a. Write and graph a system of inequalities for the number x ofnecklaces and the number y of keychains that you can make underthe given constraints.

b. Find the vertices (corner points) of the graph.

c. You sell each necklace for $10 and each keychain for $8. The revenueR is given by the equation R 5 10x 1 8y. Find the revenue for eachordered pair in part (b). Which vertex results in the maximum revenue?

Evaluate the expression.

42. 13x2 when x 5 2 (p. 8) 43. 64 4 z3 when z 5 2 (p. 8)

44. 2| 2c | 2 32 when c 5 8 (p. 64) 45. 28 1 3y 2 16 when y 5 26 (p. 88)

46. 7 2 8w}

11w when w 5 5 (p. 103) 47. 21 1 Ï

}x when x 5 144 (p. 110)

Use any method to solve the linear system. (pp. 427, 435, 444, 451)

48. y 5 4x 2 1 49. y 5 22x 2 6 50. x 1 2y 5 21y 5 28x 1 23 y 5 25x 2 12 2x 5 y 2 2

51. 4x 1 y 5 0 52. 2x 2 y 5 25 53. 3x 1 2y 5 22x 1 y 5 5 y 5 2x 2 5 23x 1 y 5 211

MIXED REVIEW

PREVIEW


Lessons 7.5–7.6

STATE TEST PRACTICEclasszone.com

1. MULTI-STEP PROBLEM A minimum of600 bricks and 12 bags of sand are neededfor a construction job. Each brick weighs2 pounds, and each bag of sand weighs50 pounds. The maximum weight that adelivery truck can carry is 3000 pounds.

a. Let x be the number of bricks, and lety be the number of bags of sand. Write asystem of linear inequalities that modelsthe situation.

b. Graph the system of inequalities.

c. Use the graph to determine whether700 bricks and 20 bags of sand can bedelivered in one trip.

2. MULTI-STEP PROBLEM Dana decides topaint the ceiling and the walls of a room.She spends $120 on 2 gallons of paint for theceiling and 4 gallons of paint for the walls.Then she decides to paint the ceiling and thewalls of another room using the same kinds ofpaint. She spends $60 for 1 gallon of paint forthe ceiling and 2 gallons of paint for the walls.

a. Write a system oflinear equationsthat models thesituation.

b. Is there enoughinformation givento determine thecost of one gallon ofeach type of paint?Explain.

c. A gallon of ceiling paint costs $3 morethan a gallon of wall paint. What is thecost of one gallon of each type of paint?

3. SHORT RESPONSE During a sale at a musicand video store, all CDs are priced the sameand all DVDs are priced the same. Karen buys4 CDs and 2 DVDs for $78. The next day, whilethe sale is still in progress, Karen goes backand buys 2 CDs and 1 DVD for $39. Is thereenough information to determine the cost of1 CD? Explain.

4. SHORT RESPONSE Two airport shuttles,bus A and bus B, take passengers to theairport from the same bus stop. The graphshows the distance d (in miles) traveledby each bus t hours after bus A leaves thestation. The distance from the bus stop tothe airport is 25 miles. If bus A and bus Bcontinue at the same rates, will bus B evercatch up to bus A? Explain.

5. EXTENDED RESPONSE During the summer,you want to earn at least $200 per week. Youearn $10 per hour working as a lifeguard, andyou earn $8 per hour working at a retail store.You can work at most 30 hours per week.

a. Write and graph a system of linearinequalities that models the situation.

b. If you work 5 hours per week as a lifeguardand 15 hours per week at the retail store,will you earn at least $200 per week?Explain.

c. You are scheduled to work 20 hours perweek at the retail store. What is the rangeof hours you can work as a lifeguard toearn at least $200 per week?

6. OPEN-ENDED Describe a real-world situationthat can be modeled by a system of linearinequalities. Then write and graph thesystem of inequalities.

7. GRIDDED ANSWER What is the area (insquare feet) of the triangular garden definedby the system of inequalities below?

y ≥ 0x ≥ 0

4x 1 5y ≤ 60

Time (hours)

Dis

tanc

e (m

iles)

1 2

30

20

10

00 t

d

Bus BBus A

Mixed Review of Problem Solving 473

MIXED REVIEW of Problem SolvingMIXED REVIEW of Problem Solving


Big Idea 1

Big Idea 3

7BIG IDEAS For Your Notebook

Solving Linear Systems by Graphing

The graph of a system of two linear equations tells you how many solutionsthe system has.

One solution No solution Infinitely many solutions

x

y

x

y

The lines intersect. The lines are parallel. The lines coincide.

Solving Linear Systems Using Algebra

You can use any of the following algebraic methods to solve a system of linearequations. Sometimes it is easier to use one method instead of another.

Method Procedure When to use

Substitution Solve one equation for x or y.Substitute the expression for xor y into the other equation.

When one equation isalready solved for x or y

Addition Add the equations to eliminatex or y.

When the coefficients ofone variable are opposites

Subtraction Subtract the equations toeliminate x or y.

When the coefficients ofone variable are the same

Multiplication Multiply one or both equationsby a constant so that adding orsubtracting the equations willeliminate x or y.

When no correspondingcoefficients are the sameor opposites

Solving Systems of Linear Inequalities

The graph of a system of linear inequalities is the intersection of thehalf-planes of each inequality in the system. For example, the graph ofthe system of inequalities below is the shaded region.

x ≤ 6 Inequality 1y < 2 Inequality 2

2x 1 3y ≥ 6 Inequality 3

Big Idea 2

x

y

x

y

1

1

CHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARY

Chapter Review 475

• system of linear equations, p. 427

• solution of a system of linearequations, p. 427

• consistent independent system,p. 427

• inconsistent system, p. 459

• consistent dependentsystem, p. 459

• system of linear inequalities,p. 466

• solution of a system of linearinequalities, p. 466

• graph of a system of linearinequalities, p. 466

REVIEW KEY VOCABULARY

Use the review examples and exercises below to check your understanding ofthe concepts you have learned in each lesson of Chapter 7.

REVIEW EXAMPLES AND EXERCISES

7

EXAMPLES1 and 2

on pp. 427–428for Exs. 5–7

Solve Linear Systems by Graphing pp. 427–433

E X A M P L E

Solve the linear system by graphing. Check your solution.

y 5 x 2 2 Equation 1y 5 23x 1 2 Equation 2

Graph both equations. The lines appear to intersect at(1, 21). Check the solution by substituting 1 for x and21 for y in each equation.

y 5 x 2 2 y 5 23x 1 2

21 0 1 2 2 21 0 23(1) 1 2

21 5 21 ✓ 21 5 21 ✓

EXERCISESSolve the linear system by graphing. Check your solution.

5. y 5 23x 1 1 6. y 5 3x 1 4 7. x 1 y 5 3y 5 x 2 7 y 5 22x 2 1 x 2 y 5 5

7.1

VOCABULARY EXERCISES

1. Copy and complete: A(n) ? consists of two or more linear inequalitiesin the same variables.

2. Copy and complete: A(n) ? consists of two or more linear equations inthe same variables.

3. Describe how you would graph a system of two linear inequalities.

4. Give an example of a consistent dependent system. Explain why thesystem is a consistent dependent system.

x

y

y 5 x 2 2

y 5 23x 1 2

1

3

CHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWclasszone.com• Multi-Language Glossary• Vocabulary practice


7

EXAMPLES1, 2, and 3

on pp. 435–437for Exs. 8–11

7.2 Solve Linear Systems by Substitution pp. 435–441

E X A M P L E

Solve the linear system: 3x 1 y 5 29 Equation 1y 5 5x 1 7 Equation 2

STEP 1 Substitute 5x 1 7 for y in Equation 1 and solve for x.

3x 1 y 5 29 Write Equation 1.

3x 1 5x 1 7 5 29 Substitute 5x 1 7 for y.

x 5 22 Solve for x.

STEP 2 Substitute 22 for x in Equation 2 to find the value of y.

y 5 5x 1 7 5 5(22) 1 7 5 210 1 7 5 23

c The solution is (22, 23). Check the solution by substituting 22 for xand 23 for y in each of the original equations.

EXERCISESSolve the linear system using substitution.

8. y 5 2x 2 7 9. x 1 4y 5 9 10. 2x 1 y 5 215x 1 2y 5 1 x 2 y 5 4 y 2 5x 5 6

11. ART Kara spends $16 on tubes of paint and disposable brushes for anart project. Each tube of paint costs $3, and each disposable brush costs$.50. Kara purchases twice as many brushes as tubes of paint. Find thenumber of brushes and the number of tubes of paint that she purchases.

7.3 Solve Linear Systems by Adding or Subtracting pp. 444–450

E X A M P L E

Solve the linear system: 5x 2 y 5 8 Equation 125x 1 4y 5 217 Equation 2

STEP 1 Add the equations toeliminate one variable.

STEP 2 Solve for y.

STEP 3 Substitute 23 for y in either equation and solve for x.

5x 2 y 5 8 Write Equation 1.

5x 2 (23) 5 8 Substitute 23 for y.

x 5 1 Solve for x.

c The solution is (1, 23). Check the solution by substituting 1 for xand 23 for y in each of the original equations.

5x 2 y 5 825x 1 4y 5 217 3y 5 29

y 5 23

CHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEW

Chapter Review 477

EXAMPLES1, 2, and 3

on pp. 444–445for Exs. 12–17

EXERCISESSolve the linear system using elimination.

12. x 1 2y 5 13 13. 4x 2 5y 5 14 14. x 1 7y 5 12x 2 2y 5 27 24x 1 y 5 26 22x 1 7y 5 18

15. 9x 2 2y 5 34 16. 3x 5 y 1 1 17. 4y 5 11 2 3x5x 2 2y 5 10 2x 2 y 5 9 3x 1 2y 5 25

EXAMPLES1 and 2

on pp. 451–452for Exs. 18–24

7.4 Solve Linear Systems by Multiplying First pp. 451–457

E X A M P L E

Solve the linear system: x 2 2y 5 27 Equation 1 3x 2 y 5 4 Equation 2

STEP 1 Multiply the first equation by 23.

x 2 2y 5 27 3 (23) 23x 1 6y 5 213x 2 y 5 4 3x 2 y 5 4

STEP 2 Add the equations. 5y 5 25

STEP 3 Solve for y. y 5 5

STEP 4 Substitute 5 for y in either of the original equations and solve for x.


x 2 2(5) 5 27 Substitute 5 for y.

x 5 3 Solve for x.




x 2 2y 5 27 3x – y 5 4

3 2 2(5) 0 27 3(3) 2 5 0 4

27 5 27 ✓ 4 5 4 ✓

EXERCISESSolve the linear system using elimination.

18. 2x 1 y 5 24 19. x 1 6y 5 28 20. 3x 2 5y 5 272x 2 3y 5 5 2x 2 3y 5 219 24x 1 7y 5 8

21. 8x 2 7y 5 23 22. 5x 5 3y 2 2 23. 11x 5 2y 2 16x 2 5y 5 21 3x 1 2y 5 14 3y 5 10 1 8x

24. CAR MAINTENANCE You pay $24.50 for 10 gallons of gasoline and 1 quartof oil at a gas station. Your friend pays $22 for 8 gallons of the samegasoline and 2 quarts of the same oil. Find the cost of 1 quart of oil.

classzone.comChapter Review Practice


7

EXAMPLES1, 2, and 3

on pp. 459–461for Exs. 25–27

EXAMPLES1, 2, 3, and 4

on pp. 466–468for Exs. 28–31

x

y

y ≥ x 2 3

y < 22x 1 3

1

1

7.5 Solve Special Types of Linear Systems pp. 459–465

E X A M P L E

Show that the linear system has no solution. 22x 1 y 5 23 Equation 1y 5 2x 1 1 Equation 2


The lines are parallel because they havethe same slope but different y-intercepts.Parallel lines do not intersect, so the systemhas no solution.

EXERCISESTell whether the linear system has one solution, no solution, or infinitelymany solutions. Explain.

25. x 5 2y 2 3 26. 2x 1 y 5 8 27. 4x 5 2y 1 6 1.5x 2 3y 5 0 x 1 8 5 y 4x 1 2y 5 10

7.6 Solve Systems of Linear Inequalities pp. 466–472

E X A M P L E

Graph the system of linear inequalities. y < 22x 1 3 Inequality 1y ≥ x 2 3 Inequality 2

The graph of y < 22x 1 3 is the half-plane belowthe dashed line y 5 22x 1 3.

The graph of y ≥ x 2 3 is the half-plane on andabove the solid line y 5 x 2 3.

The graph of the system is the intersection ofthe two half-planes shown as the darker shadeof blue.

EXERCISESGraph the system of linear inequalities.

28. y < x 1 3 29. y ≤ 2x 2 2 30. y ≥ 0y > 23x 2 2 y > 4x 1 1 x ≤ 2

y < x 1 4

31. MOVIE COSTS You receive a $40 gift card to a movie theater. A ticket to amatinee movie costs $5, and a ticket to an evening movie costs $8. Writeand graph a system of inequalities for the number of tickets you canpurchase using the gift card.

x

y

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

2

12

12

22

CHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEW

Chapter Test 479

7Solve the linear system by graphing. Check your solution.

1. 3x 2 y 5 26 2. 22x 1 y 5 5 3. y 5 4x 1 4x 1 y 5 2 x 1 y 5 21 3x 1 2y 5 12

4. 5x 2 4y 5 20 5. x 1 3y 5 9 6. 2x 1 7y 5 14x 1 2y 5 4 2x 2 y 5 4 5x 1 7y 5 27

Solve the linear system using substitution.

7. y 5 5x 2 7 8. x 5 y 2 11 9. 3x 1 y 5 21924x 1 y 5 21 x 2 3y 5 1 x 2 y 5 7

10. 15x 1 y 5 70 11. 3y 1 x 5 17 12. 0.5x 1 y 5 93x 2 2y 5 28 x 1 y 5 8 1.6x 1 0.2y 5 13

Solve the linear system using elimination.

13. 8x 1 3y 5 29 14. x 2 5y 5 23 15. 4x 1 y 5 1728x 1 y 5 29 3x 2 5y 5 11 7y 5 4x 2 9

16. 3x 1 2y 5 25 17. 3y 5 x 1 5 18. 6x 2 5y 5 9x 2 y 5 10 23x 1 8y 5 8 9x 2 7y 5 15

Tell whether the linear system has one solution, no solution, or infinitelymany solutions.

19. 15x 2 3y 5 12 20. 4x 2 y 5 24 21. 212x 1 3y 5 18y 5 5x 2 4 28x 1 2y 5 2 4x 1 y 5 26

22. 6x 2 7y 5 5 23. 3x 2 4y 5 24 24. 10x 2 2y 5 14212x 1 14y 5 10 3x 1 4y 5 24 15x 2 3y 5 21

Graph the system of linear inequalities.

25. y < 2x 1 2 26. y ≤ 3x 2 2 27. y ≤ 3y ≥ 2x 2 1 y > x 1 4 x > 21

y > 3x 2 3

28. TRUCK RENTALS Carrie and Dave each rent the same size moving truckfor one day. They pay a fee of x dollars for the truck and y dollars per milethey drive. Carrie drives 150 miles and pays $215. Dave drives 120 milesand pays $176. Find the amount of the fee and the cost per mile.

29. GEOMETRY The rectangle has a perimeter P of58 inches. The length l is one more than 3 times thewidth w. Write and solve a system of linear equationsto find the length and width of the rectangle.

30. COMMUNITY SERVICE A town committee has a budget of $75 to spend onsnacks for the volunteers participating in a clean-up day. The committeechairperson decides to purchase granola bars and at least 50 bottles ofwater. Granola bars cost $.50 each, and bottles of water cost $.75 each.Write and graph a system of linear inequalities for the number of bottlesof water and the number of granola bars that can be purchased.

l

wP 5 58 in.

CHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TEST


MULTIPLE CHOICE QUESTIONS

If you have difficulty solving a multiple choice problem directly, you may be able to use another approach to eliminate incorrect answer choices and obtain the correct answer.

Which ordered pair is the solution of the linear system y 5 1}2

xand 2x 1 3y 5 27?

A (2, 1) B (1, 23) C (22, 21) D (4, 2)

PRO B L E M 1

★ Standardized TEST PREPARATION7

Method 1SOLVE DIRECTLY Use substitution to solve the linear system.

STEP 1 Substitute 1}2

x for y in the equation

2x 1 3y 5 27 and solve for x.

2x 1 3y 5 27

2x 1 3 11}2

x 2 5 27

2x 1 3}2

x 5 27

7}2

x 5 27

x 5 22

STEP 2 Substitute 22 for x in y 5 1}2

x to find the value of y.

y 5 1}2

x

5 1}2

(22)

5 21

The solution of the system is (22, 21).

The correct answer is C. A B C D

Method 2ELIMINATE CHOICES Substitute the values given in each answer choice for x and y in both equations.

Choice A: (2, 1)Substitute 2 for x and 1 for y.

y 5 1}2

x 2x 1 3y 5 27

1 0 1}2

(2) 2(2) 1 3(1) 0 27

1 5 1 ✓ 7 5 27 ✗

Choice B: (1, 23)Substitute 1 for x and 23 for y.

y 5 1}2

x

23 0 1}2

(1)

23 5 1}2

✗

Choice C: (22, 21)Substitute 22 for x and 21 for y.

y 5 1}2

x 2x 1 3y 5 27

21 0 1}2

(22) 2(22) 1 3(21) 0 27

21 5 21 ✓ 27 5 27 ✓


Standardized Test Preparation 481

PRACTICEExplain why you can eliminate the highlighted answer choice.

1. The sum of two numbers is –27. One number is twice the other. What are the numbers?

A 9 and 18 B 23 and 24 C 218 and 29 D 214 and 213

2. Which ordered pair is a solution of the linear system 5x 1 2y 5 211

and x 5 21}2 y 2 4?

A 127}6, 2

17}3 2 B 1223

}3 ,

1}3 2 C (23, 2) D (23, 213)

3. Long-sleeve and short-sleeve T-shirts can be purchased at a concert. A long-sleeve T-shirt costs $25 and a short-sleeve T-shirt costs $15. During a concert, the T-shirt vendor collects $8415 from the sale of 441 T-shirts. How many short-sleeve T-shirts were sold?

A 100 B 180 C 261 D 441

The sum of two numbers is 21, and the difference of the two numbers is 5. What are the numbers?

A 25 and 4 B 1 and 6 C 2 and 23 D 22 and 3

PRO B L E M 2

Method 1SOLVE DIRECTLY Write and solve a system of equations for the numbers.

STEP 1 Write a system of equations. Let x and ybe the numbers.

x 1 y 5 21 Equation 1x 2 y 5 5 Equation 2

STEP 2 Add the equations to eliminate one variable. Then find the value of the other variable.

x 1 y 5 21x 2 y 5 5

2x 5 4, so x 5 2

STEP 3 Substitute 2 for x in Equation 1 and solve for y.

2 1 y 5 21, so y 5 23


Method 2ELIMINATE CHOICES Find the sum and difference of each pair of numbers. Because the difference is positive, be sure to subtract the lesser number from the greater number.

Choice A: 25 and 4

Sum: 25 1 4 5 21 ✓

Difference: 25 2 4 5 29 ✗

Choice B: 1 and 6

Sum: 1 1 6 5 7 ✗

Choice C: 2 and 23

Sum: 2 1 (23) 5 21 ✓

Difference: 2 2 (23) 5 5 ✓



1. Which ordered pair is the solution of the

linear system y 5 1}2

x 1 1 and y 5 3}2

x 1 4?

A 13, 5}2

2 B 123, 2 1}2

2

C 13}2

, 7}4

2 D (0, 1)

2. How many solutions does the linear system 3x 1 5y 5 8 and 3x 1 5y 5 1 have?

A 0 B 1

C 2 D Infinitely many

3. The sum of two numbers is 23, and the difference of the two numbers is 11. What are the numbers?

A 4 and 7 B 23 and 8

C 3 and 14 D 27 and 4

4. Which ordered pair is the solution of the system of linear equations whose graph is shown?

A (3, 21) B (0, 0)

C (0, 2) D (23, 1)

5. Which ordered pair is the solution of the

linear system 3x 1 y 5 21 and y 5 21}2

x 2 7}2

?

A 15}2

, 217}2 2 B (1, 24)

C (21, 23) D (0, 21)

6. Which ordered pair is a solution of the system x 1 2y ≤ 22 and y ≤ 23x 1 4?

A (0, 0) B (2, 22)

C (22, 2) D (5, 24)

7. How many solutions does the system of linear equations whose graph is shown have?

A 0 B 1

C 2 D Infinitely many

8. At a bakery, one customer pays $5.67 for 3 bagels and 4 muffins. Another customer pays $6.70 for 5 bagels and 3 muffins. Which system of equations can be used to determine the cost x (in dollars) of one bagel and the cost y (in dollars) of one muffin at the bakery?

A x 1 y 5 7 B y 5 3x 1 5.67 x 1 y 5 8 y 5 5x 1 6.7

C 3x 1 4y 5 6.7 D 3x 1 4y 5 5.675x 1 3y 5 5.67 5x 1 3y 5 6.7

9. The perimeter P (in feet) of each of the two rectangles below is given. What are the values of l and w?

A l5 7 and w 5 5

B l5 8 and w 5 4

C l5 11 and w 5 10

D l5 12 and w 5 9

MULTIPLE CHOICE

★ Standardized TEST PRACTICE7

x

y

1

4

x

y

1

1

l ft (l1 4) ft

w ft

2w ft

P 5 24 ft

P 5 42 ft

Standardized Test Practice 483

10. What is the x-coordinate of the solution ofthe system whose graph is shown?

11. What is the x-coordinate of the solution of

the system y 5 1}3

x 1 8 and 2x 2 y 5 2?

12. A science museum charges one amount foradmission for an adult and a lesser amountfor admission for a student. Admission tothe museum for 28 students and 5 adultscosts $284. Admission for 40 students and10 adults costs $440. What is the admissioncost (in dollars) for one student?

13. Is it possible to find a value for c so that thelinear system below has exactly one solution?Explain.


y 5 25}3x 1 c Equation 2

14. A rental car agency charges x dollars per dayplus y dollars per mile to rent any of the mid-sized cars at the agency. The total costs fortwo customers are shown below.

Customer Time(days)

Distance(miles)

Cost(dollars)

Jackson 3 150 217.50

Bree 2 112 148.00

How much will it cost to rent a mid-sized carfor 5 days and drive 250 miles? Explain.

x

y

22

1

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GRIDDED ANSWER

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15. A baseball player’s batting average is the number of hits the player hasdivided by the number of at-bats. At the beginning of a game, a player hasa batting average of .360. During the game, the player gets 3 hits during5 at-bats, and his batting average changes to .375.

a. Write a system of linear equations that represents the situation.

b. How many at-bats has the player had so far this season?

c. Another player on the team has a batting average of .240 at thebeginning of the same game. During the game, he gets 3 hits during5 at-bats, and his batting average changes to .300. Has this player hadmore at-bats so far this season than the other player? Explain.

16. A gardener combines x fluid ounces of a 20% liquid fertilizer and80% water mix with y fluid ounces of a 5% liquid fertilizer and95% water mix to make 30 fluid ounces of a 10% liquid fertilizerand 90% water mix.

a. Write a system of linear equations that represents the situation.

b. Solve the system from part (a).

c. Suppose the gardener combines pure (100%) water and the 20% liquidfertilizer and 80% water mix to make the 30 fluid ounces of the10% liquid fertilizer and 90% water mix. Is more of the 20% liquidfertilizer and 80% water mix used in this mix than in the original mix?Explain.

484 Cumulative Review: Chapters 1–7

Evaluate the expression.

1. 25 p 2 2 4 4 2 (p. 8) 2. 24 4 6 1 (9 2 6) (p. 8) 3. 5[(6 2 2)2 2 5] (p. 8)

4. Ï}

144 (p. 110) 5. 2Ï}

2500 (p. 110) 6. 6Ï}

400 (p. 110)

Check whether the given number is a solution of the equation or inequality. (p. 21)

7. 7 1 3x 5 16; 3 8. 21y 1 1 5 1; 0 9. 20 2 12h 5 12; 1

10. g 2 3 > 2; 5 11. 10 ≥ 4 2 x; 0 12. 30 2 4p ≥ 5; 6

Simplify the expression.

13. 5(y 2 1) 1 4 (p. 96) 14. 12w 1 (w 2 2)3 (p. 96) 15. (g 2 1)(24) 1 3g (p. 96)

16. 10h 2 25}

5 (p. 103) 17. 21 2 4x

}27

(p. 103) 18. 32 2 20m}2 (p. 103)

Solve the equation.

19. x 2 8 5 21 (p. 134) 20. 21 5 x 1 3 (p. 134) 21. 6x 5 242 (p. 134)

22. x}3 5 8 (p. 134) 23. 5 2 2x 5 11 (p. 141) 24. 2}3

x 2 3 5 17 (p. 141)

25. 3(x 2 2) 5 215 (p. 148) 26. 3(5x 2 7) 5 5x 2 1 (p. 154) 27. 27(2x 2 10) 5 4x 2 10 (p. 154)

Graph the equation.

28. x 1 2y 5 28 (p. 225) 29. 22x 1 5y 5 210 (p. 225) 30. 3x 2 4y 5 12 (p. 225)

31. y 5 3x 2 7 (p. 244) 32. y 5 x 1 6 (p. 244) 33. y 5 21}3x (p. 253)

Write an equation of the line in slope-intercept form with the given slope and y-intercept. (p. 283)

34. slope: 5 35. slope: 21 36. slope: 27y-intercept: 21 y-intercept: 3 y-intercept: 0

Write an equation in point-slope form of the line that passes through the given points. (p. 302)

37. (1, 210), (25, 2) 38. (4, 7), (24, 3) 39. (29, 22), (26, 8)

40. (21, 1), (1, 23) 41. (2, 4), (8, 2) 42. (26, 1), (3, 25)

Solve the inequality. Then graph your solution.

43. x 2 9 < 213 (p. 356) 44. 8 ≤ x 1 7 (p. 356) 45. 8x ≥ 56 (p. 363)

46. x}24

> 7 (p. 363) 47. 1 2 2x < 11 (p. 369) 48. 8 > 23x 2 1 (p. 369)

49. 4x 2 10 ≤ 7x 1 8 (p. 369) 50. 7x 2 5 < 6x 2 4 (p. 369) 51. 24 < 3x 2 1 < 5(p. 380)

52. 3 ≤ 9 2 2x ≤ 15 (p. 380) 53. ⏐3x⏐ < 15 (p. 398) 54. ⏐4x 2 2⏐ ≥ 18 (p. 398)

Chapters 1–7111111––777777CUMULATIVE REVIEWCUMULATIVECUMULATIVECUMULATIVECUMULATIVECUMULATIVECUMULATIVE REVIEWREVIEWREVIEWREVIEWREVIEWREVIEW

Cumulative Review: Chapters 1–7 485

Solve the linear system using elimination. (p. 451)

55. 4x 1 y 5 8 56. 3x 2 5y 5 5 57. 12x 1 7y 5 35x 2 2y 5 23 x 2 5y 5 24 8x 1 5y 5 1

58. ART PROJECT You are making a tile mosaic on the rectangular tabletop shown. A bag of porcelain tiles costs $3.95 and covers 36 square inches. How much will it cost to buy enough tiles to cover the tabletop? (p. 28)

59. FOOD The table shows the changes in the price for a dozen grade A, large eggs over 4 years. Find the average yearly change to the nearest cent in the price for a dozen grade A, large eggs during the period 1999–2002. (p. 103)

Year 1999 2000 2001 2002

Change in price for a dozen grade A, large eggs (dollars)

20.17 0.04 20.03 0.25

60. HONEY PRODUCTION Honeybees visit about 2,000,000 flowers to make 16 ounces of honey. About how many flowers do honeybees visit to make 6 ounces of honey? (p. 168)

61. MUSIC The table shows the price p (in dollars) for various lengths of speaker cable. (p. 253)

Length, l (feet) 3 5 12 15

Price, p (dollars) 7.50 12.50 30.00 37.50

a. Explain why p varies directly with l.

b. Write a direct variation equation that relates l and p.

62. CURRENCY The table shows the exchange rate between the currency of Bolivia (bolivianos) and U.S. dollars from 1998 to 2003. (p. 335)

Year 1998 1999 2000 2001 2002 2003

Bolivianos per U.S. dollar 5.51 5.81 6.18 6.61 7.17 7.66

a. Find an equation that models the bolivianos per U.S. dollar as a function of the number of years since 1998.

b. If the trend continues, predict the number of bolivianos per U.S. dollar in 2010.

63. BATTERIES A manufacturer of nickel-cadmium batteries recommends storing the batteries at temperatures ranging from 220°C to 45°C. Use an inequality to describe the temperatures (in degrees Fahrenheit) at which the batteries can be stored. (p. 380)

30 in.

24 in.

7and inequalities - wikispaces7... · systems of equations 7 and inequalities ... chapter 7:...

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