Algebra

C h

a p

t e

r

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 457

7 Systems of Linear Equations What determines the prices of the products that you buy? Why do prices of some

products go down while the prices of others go up? Economists theorize that

prices result from the collective decisions of consumers and producers. Ideally, the

demand or quantity purchased by consumers depends only on the price,and price

is a function of the supply. Theoretically, if the demand is greater than the supply,

then prices rise and manufacturers produce more to meet the demand. As the

supply of goods increases, the price comes down. The price at which the supply is

equal to the demand is called the equilibrium price.

Q ua

nt it

y (p

ou nd

s/ da

y)

1000

800

600

400

Point of equilibrium

Demand y 150x 900

Supply y 200x 60

However, what happens in the real

7.1

7.2

7.3

7.4

world does not always match the theory. The Graphing Method Manufacturers cannot always control the supply,

The Substitution Method

The Addition Method

and factors other than price can affect a con­

sumer’s decision to buy. For example, droughts

in Brazil decreased the supply of coffee and drove

coffee prices up. Floods in California did the same Systems of Linear Equations in Three to the prices of produce. With one of the most Variables abundant wheat crops ever in 1994, cattle gained 200

weight more quickly, increasingthesupplyofcat­

tle ready for market.With supply going up, prices

went down. Decreased demand for beef in Japan

and Mexico drove the price of beef down further.

With lower prices, consumers should be buying

more beef, but increased competition from

chicken and pork products, as well as health con­

cerns,have kept consumer demand low.

The two functions that govern supply

and demand form a system of equations.

In this chapter you will learn how to solve

systems of equations.

In Exercise 65 of Section 7.2 you will see an example of

supply and demand equations for ground beef.

0 1 2 3 4 5 6 Price of ground beef

(dollars/pound)

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 458

458 Chapter 7 Systems of Linear Equations 7-2

7.1 The Graphing Method

You studied linear equations in two variables in Chapter 3. In this section, you will learn to solve systems of linear equations in two variables and use systems to solve problems.

In This Section

U1V Solving a System by Graphing

U2V Types of Systems

U3V Applications

U1V Solving a System by Graphing Consider the linear equation y = 2x - 1. The graph of this equation is a straight line, and every point on the line is a solution to the equation. Now consider a second linear equation, x + y = 2. The graph of this equation is also a straight line, and every point on the line is a solution to this equation. Taken together, the pair of equations

y = 2x - 1

x + y = 2

is called a system of equations. A point that satisfies both equations is called a solution to the system.

E X A M P L E 1 A solution to a system Determine whether the point (-1, 3) is a solution to each system of equations.

a) 3x - y = -6 b) y = 2x - 1 x + 2y = 5 x + y = 2

Solution a) If we let x = -1 and y = 3 in both equations of the system, we get the following

equations:

3(-1) - 3 = -6 Correct

-1 + 2(3) = 5 Correct

Because both of these equations are correct, (-1, 3) is a solution to the system.

b) If we let x = -1 and y = 3 in both equations of the system, we get the following equations:

3 = 2(-1) - 1 Incorrect

-1 + 3 = 2 Correct

Because the first equation is not satisfied by (-1, 3), the point (-1, 3) is not a solution to the system.

U Calculator Close-Up V

Solve both equations in Example 1(a) for y to get y = 3x + 6 and y = (5 - x)/2. The graphs show that (-1, 3) is on both lines.

For Example 1(b), graph y = 2x - 1 and y = 2 - x to see that (-1, 3) is on one line but not the other.

10

10

1010

10 10

10 Now do Exercises 1–8

If we graph each equation of a system on the same coordinate plane, then we may be able to see the points that they have in common. Any point that is on both graphs 10 is a solution to the system.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 459

7-3 7.1 The Graphing Method 459

E X A M P L E 2 A system with only one solution Solve the system by graphing:

y = x + 2

x + y = 4

Solution First write the equations in slope-intercept form:

y = x + 2

y = -x + 4

Use the y-intercept and the slope to graph each line. The graph of the system is shown in Fig. 7.1. From the graph it appears that these lines intersect at (1, 3). To be certain, we can check that (1, 3) satisfies both equations. Let x = 1 and y = 3 in y = x + 2 to get

3 = 1 + 2.

Let x = 1 and y = 3 in x + y = 4 to get

1 + 3 = 4.

Because (1, 3) satisfies both equations, the solution set to the system is {(1, 3)}.

Figure 7.1

y

x13 1

1

32

3

5

2

1

?

y x 2

y x 4

U Calculator Close-Up V

To check Example 2, graph

y1 = x + 2

and

y2 = -x + 4.

From the CALC menu, choose intersect to have the calculator locate the point of intersection of the two lines. After choosing intersect, you must indicate which two lines you want to intersect and then guess the point of intersection.

10

10

10

10

Now do Exercises 9–16

E X A M P L E 3 A system with exactly one solution Solve the system by graphing:

x - y = 6

2x + y = 6

Solution We can graph these equations using their x- and y-intercepts. The intercepts for x - y = 6 are (6, 0) and (0, -6). The intercepts for 2x + y = 6 are (3, 0) and (0, 6). Draw the graphs

In Example 3, we graph the lines using the x- and y-intercepts.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 460

460 Chapter 7 Systems of Linear Equations 7-4

through the intercepts as shown in Fig. 7.2. The lines appear to cross at (4, -2). To be certain, check (4, -2) in both equations:

x - y = 6 2x + y = 6

4 - (-2) = 6 Correct 2 · 4 + (-2) = 6 Correct

Because both of the equations are correct, (4, -2) is the solution to the system. The solu­ tion set is {(4, -2)}.

1 2

4

6

2

3 4 6 7 2

4

1 x

y

2x y 6

x y 6 Now do Exercises 17–24

Figure 7.2

E X A M P L E 4 A system with infinitely many solutions Solve the system by graphing:

4x - 2y = 6

y - 2x = -3

Solution Rewrite both equations in slope-intercept form for easy graphing:

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

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

y = 2x - 3

By writing the equations in slope-intercept form, we discover that they are identical. So the equations have the same graph, which is shown in Fig. 7.3. So any point on that line satisfies both of the equations, and there are infinitely many solutions to the system. The solution set consists of all points on the line y = 2x - 3, which is written in set notation as

{(x, y) I y = 2x - 3}.

3

3

y

x

2

2 4

1

1

4x 2y 6 y 2x 3

2 1

3

2

1

Figure 7.3

Now do Exercises 33–36

In Example 4 we read {(x, y) I y = 2x - 3} as “the set of ordered pairs (x, y) such that y = 2x - 3.” Note that we could have used 4x - 2y = 6 or y - 2x = -3 in place of y = 2x - 3 in set notation since these three equations are equivalent. We usually choose the simplest equation for set notation.

E X A M P L E 5 A system with no solution Solve the system by graphing:

3y = 2x - 6

2x - 3y = 3

Solution Write each equation in slope-intercept form to get the following system:

2 y = ?? x - 2

3

2 y = ?? x - 1

3

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 461

7-5 7.1 The Graphing Method 461

Each line has slope ? 2 3

?, but they have different y-intercepts. Their graphs are shown in Fig. 7.4. Because these two lines have equal slopes, they are parallel. There is no point of intersection and no solution to the system.

Now do Exercises 37–40

3

y

x1

3

2

1

4

2 2x 3y 3

3y 2x 6

3

U2V Types of Systems A system of equations that has at least one solution is consistent (Examples 2, 3, and 4). A system with no solutions is inconsistent (Example 5). There are two types of con­ sistent systems. A consistent system with exactly one solution is independent (Examples 2 and 3) and a consistent system with infinitely many solutions isFigure 7.4 dependent (Example 4). These ideas are summarized in Fig. 7.5.

You can classify a system as independent, dependent, or inconsistent by examin­ ing the slope-intercept form of each equation, as shown in Example 6.

Consistent systems: Independent Exactly one solution

1

1

1 2 3 4 5

2

3

4

5

1

y

x

2x y 1

x y 5

Dependent Infinitely many solutions

1

1

1 2 3 4 5

2

3

4

5

1

y

x

x y 5

2x 2y 10

Inconsistent system:

No solution

1

1

1 2 3 4 5

2

3

4

5

1

y

x

x y 5

x y 3

Figure 7.5

E X A M P L E 6 Types of systems Determine whether each system is independent, dependent, or inconsistent.

a) y = 3x - 5 b) y = 2x + 3 c) y = 5x - 1 y = 3x + 2 y = -2x + 5 2y - 10x = -2

Solution a) Since y = 3x - 5 and y = 3x + 2 have the same slope and different y-intercepts, the

two lines are parallel. There is no point of intersection. The system is inconsistent.

b) Since y = 2x + 3 and y = -2x + 5 have different slopes, they are not parallel. These two lines intersect at a single point. The system is independent.

c) First rewrite the second equation in slope-intercept form:

2y - 10x = -2 2y = 10x - 2 y = 5x - 1

Since the first equation is also y = 5x - 1, these are two different-looking equations for the same line. So every point on that line satisfies both equations. The system is dependent.

Now do Exercises 41–54

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 462

462 Chapter 7 Systems of Linear Equations 7-6

U Calculator Close-Up V

U3V Applications In a simple economic model, both supply and demand depend only on price. Supply is the quantity of an item that producers are willing to make or supply. Demand is the quantity consumers will purchase. As the price increases, producers increase the sup­ ply to take advantage of rising prices. However, as the price increases, consumer demand decreases. The equilibrium price is the price at which supply equals demand.

E X A M P L E 7 Supply and demand Monthly demand for Greeny Babies (small toy frogs) is given by the equation y = 8000 - 400x, while monthly supply is given by the equation y = 400x, where x is the price in dollars. Graph the two equations, and find the equilibrium price and the demand at the equilibrium price.

Solution The graph of y = 8000 - 400x goes through (0, 8000) and (20, 0). The graph of y = 400x goes through (0, 0) and (20, 8000). The two lines cross at (10, 4000) as shown in Fig. 7.6. So the equilibrium price is $10, and the monthly demand is 4000 Greeny Babies.

Now do Exercises 69–72

0 5 10 15 20

2000

4000

6000

8000

Price

N um

be r

of to

ys

(20, 8000)

Demand

x

y

Equilibrium (10, 4000) Supply

(0, 8000)

(20, 0)(0, 0)

Figure 7.6

With a graphing calculator, you can graph both equations of a system in a single view­ ing window. The TRACE feature can then be used to estimate the solution to an indepen­ dent system. You could also use ZOOM to

10

10 10

10

“blow up” the intersection and get more accuracy. Many calculators have an intersect feature, which can find a point of intersection. First graph y1 = 2x - 1 and y2 = 2 - x.

From the CALC menu choose intersect.

Verify the curves (or lines) that you want to intersect by pressing ENTER. After you make a guess as to the intersection by posi­ tioning the cursor or entering a number, the calculator will find the intersection.

10

10 10

10

dug84356_ch07a.qxd 9/18/10 8:53 AM Page 463

7-7 7.1 The Graphing Method 463

Warm-Ups ▼

Fill in the blank. 1. A pair of equations is a of equations.

2. An ordered pair that satisfies both equations is a to the system.

3. A system of equations that has at least one solution is .

4. The solution to an linear system in two variables is the point of intersection of the two lines.

5. A consistent system with infinitely many solutions is .

6. If the two lines are , then there is no solution to the linear system.

7. If the two lines , then there are infinitely many solutions to the linear system.

True or false? 8. The point (1, 2) satisfies 2x + y = 4. 9. The point (1, 2) satisfies 2x + y = 4 and 3x - y = 6.

10. The point (2, 3) satisfies 4x - y = 5 and 4x - y = -5.

11. If two distinct lines in a plane are not parallel, then they intersect at exactly one point.

12. No ordered pair satisfies y = 3x - 5 and y = 3x + 1.

7 .1Exercises

U Study Tips V • Working problems 1 hour per day every day of the week is better than working problems for 7 hours on one day of the week. Spread

out your study time. Avoid long study sessions. • No two students learn in exactly the same way or at the same speed. Figure out what works for you.

U1V Solving a System by Graphing

Which of the given points is a solution to the given system? See Example 1.

1. 2x + y = 4 (6, 1), (3, -2), (2, 4)

x - y = 5

2. 2x - 3y = -5 (-1, 1), (3, 4), (2, 3)

y = x + 1

3. 6x - 2y = 4 (0, -2), (2, 4), (3, 7)

y = 3x - 2

4. y = -2x + 5 (9, -13), (-1, 7), (0, 5)

4x + 2y = 10

5. 2x - y = 3 (3, 3), (5, 7), (7, 11)

2x - y = 2

6. y = x + 5 (1, -2), (3, 0), (6, 3)

y = x - 3

Use the given graph to find an ordered pair that satisfies each system of equations. Check that your answer satisfies both equations of each system.

7. y = 3x + 9 8. x - 2y = 5

2 2x + 3y = 5 y = - ?? x + 1

3

x 1

2

1 3 5

2

y

x 2y 5

y x 12 — 3

1

y

x 1

1

3

4

1

2x 3y 5

y 3x 9

2 4 5

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 464

464 Chapter 7 Systems of Linear Equations 7-8

Solve each system by graphing. See Examples 2 and 3. U2V Types of Systems

9. y = 2x 10. y = 3x Determine whether each system is independent, dependent, or y = -x + 6 y = -x + 4 inconsistent. See Example 6.

11. 3x - y = 1 12. 2x + y = 3 1 2y - 3x = 1 x + y = 1 41. y = ??x + 3 42. y = -3x - 60 2

13. x - y = 5 14. y + 4x = 10 1 1 y = ??x - 5 y = ??x - 60x + y = -5 2x - y = 2 2 3

15. 2y + x = 4 16. 2x + y = -1 43. y = 4x + 3 44. y = 5x - 4 2x - y = -7 x + y = -2 y = 3 + 4x y = 4 + 5x

17. y = x 18. x = 2y 1 45. y = ??x + 3 46. y = -x - 1 x + y = 0 0 = 9x - y 2

19. y = 2x - 1 20. y = x - 1 y = -3x - 1 y = -1 - x x - 2y = -4 2x - y = 0 47. 2x - 3y = 5 48. x + y = 1

21. x - y = 2 22. x - y = -1 2x - 3y = 7 2x + 2y = 2 x + 3y = 6 3x - y = 3

Use the following graph to determine whether the systems in 23. x - 2y = -8 24. x + 3y = 9 Exercises 49–54 are independent, dependent, or inconsistent. 3x - 2y = -12 2x + 3y = 12

Solve each system by graphing both equations on a graphing calculator and using the intersection feature of the calculator to find the point of intersection.

25. y = x + 5 26. y = 2x + 1 y = 9 - x y = 5 - 2x

27. y = 3x - 18 28. y = -x + 26 y = 32 - 2x y = 2x - 34

x -1

-2

1 2

2

1

y

-2-1

y = x + 2

y = x - 2 x + y = 2

29. x + y = 12 30. x - y = -10 3x + 2y = 14 x - 4y = 20 49. y = x - 2 50. y = x - 2

31. x + 5y = -1 32. x + y = 0.6 y = x + 2 x - y = 2

x - 5y = 2 2y + 3x = -0.5 51. y = x + 2 52. y = x - 2 x + y = 2 x + y = 2

53. y = x + 2 54. x - y = 2 x - y = 2 x + y = 2

Solve each system by graphing. See Examples 4 and 5. Solve each system by graphing. Indicate whether each system is33. x - y = 3 independent, dependent, or inconsistent. See Examples 2–6.3x = 3y + 9 55. x - y = 334. 2x + y = 3

3x = y + 56x - 9 = -3y 56. 3x + 2y = 6

35. 4y - 2x = -16 2x - y = 4 x - 2y = 8 57. x - y = 5

36. x - y = 0 x - y = 8 5x = 5y 58. y + 3x = 6

37. x - y = 3 y - 5 = -3x

3x = 3y + 12 1 59. y = ?? x + 2 338. 2y = -3x + 6 1

2y = -3x - 2 y = -?? x 3

39. x + y = 4 60. y - 4x = 4 2y = -2x + 6 y + 4x = -4

40. y = 3x - 5 61. x - y = 1 y - 3x = 0 -2y = -2x + 2

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 465

7-9

1 62. x = ??y

3 y = 3x

63. x - y = -1 1

y = ?? x - 1 2

64. y = -3x + 1 2 - 2y = 6x

The graphs of the following systems are given in (a) through (d). Match each system with the correct graph.

65. 5x + 4y = 7 66. 3x - 5y = -9 x - 3y = 9 5x - 6y = -8

67. 4x - 5y = -2 68. 4x + 5y = -2 3y - x = -3 4y - x = 11

a) b)

c) d) y

x1 2 3 4

3

4

2

1 1

2

y

x12 1 3 4

4

1

2

1

y

x

3

2

1

23

3

2

4

y

x123 1 2

3

3

1

2

2

1

4

U3V Applications Solve each problem by using the graphing method. See Example 7.

69. Competing pizzas. Mamma’s Pizza charges $10 plus $2 per topping for a deep dish pizza. Papa’s Pizza charges $5 plus $3 per topping for a similar pizza. The equations C = 2n + 10 and C = 3n + 5 express the cost C at each restaurant in terms of the number of toppings n.

a) Solve this system of equations by examining the accompanying graph.

b) Interpret the solution.

7.1 The Graphing Method 465

Figure for Exercise 69

Number of toppings

10

C os

t i n

do ll

ar s

20

30

2 4 6 8 10

C 3n 5

C 2n 10

70. Equilibrium price. A manufacturer plans to supply y units of its model 1020P CD player per month when the retail price is p dollars per player, where y = 6p + 100. Consumer studies show that consumer demand for the model 1020P is y units per month, where y = -3p + 910.

a) Fill in the missing entries in the following table.

Price Supply Demand

$ 0

50

100

300

b) Use the data in part (a) to graph both linear equations on the same coordinate system.

c) What is the price at which the supply is equal to the demand, the equilibrium price?

71. Cost of two copiers. An office manager figures the total cost in dollars for a certain used Xerox copier is given by C = 800 + 0.05x, where x is the number of copies made. She is also considering a used Panasonic copier for which the total cost is C = 500 + 0.07x.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 466

466 Chapter 7 Systems of Linear Equations 7-10

a) Fill in the missing entries in the following table.

Number of Copies

Cost Xerox

Cost Panasonic

0

5000

10,000

20,000

b) Use the data from part (a) to graph both equations on the same coordinate system.

c) For what number of copies is the total cost the same for either copier?

d) If she plans to buy another copier before 10,000 copies are made, then which copier is cheaper?

72. Flat tax proposals. Representative Schneider has proposed

Getting More Involved

73. Discussion

If both (-1, 3) and (2, 7) satisfy a system of two linear equations, then what can you say about the system?

74. Cooperative learning

Working in groups, write an independent system of two linear equations whose solution is (3, 5). Each group should then give its system to another group to solve.

75. Cooperative learning

Working in groups, write an inconsistent system of linear equations such that (-2, 3) satisfies one equation and (1, 4) satisfies the other. Each group should then give its system to another group to solve.

76. Cooperative learning

Suppose that 2x + 3y = 6 is one equation of a system. Find the second equation given that (4, 8) satisfies the second equation and the system is inconsistent.

a flat income tax of 15% on earnings in excess of $10,000. Graphing Calculator Exercises Under his proposal the tax T for a person earning E dollars is given by T = 0.15(E - 10,000). Representative Humphries has proposed that the income tax should be 20% on earnings in excess of $20,000, or T = 0.20(E - 20,000). Graph both linear equations on the same coordinate system. For what earnings would you pay the same amount of income tax under either plan? Under which plan does a rich person pay less income tax?

Solve each system by graphing each pair of equations on a graphing calculator and using the calculator to estimate the point of intersection. Give the coordinates of the intersection to the nearest tenth.

77. y = 2.5x - 6.2

y = -1.3x + 8.1

78. y = 305x + 200

y = -201x - 999

79. 2.2x - 3.1y = 3.4

5.4x + 6.2y = 7.3

80. 34x - 277y = 1

402x + 306y = 12,000

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 467

7-11 7.2 The Substitution Method 467

7.2 The Substitution Method

Solving a system by graphing is certainly limited by the accuracy of the graph. If the lines intersect at a point whose coordinates are not integers, then it is difficult to identify the solution from a graph. In this section we introduce a method for solving systems of linear equations in two variables that does not depend on a graph and is totally accurate.

In This Section

U1V Solving a System by Substitution

U2V Dependent and Inconsistent Systems

U3V Applications

U1V Solving a System by Substitution To solve a system by substitution we replace a variable in one equation by an equivalent expression for that variable (obtained from the other equation). The result should be an equation in only one variable, which we can solve by the usual techniques.

E X A M P L E 1 Solving a system by substitution Solve:

3x + 4y = 5

x = y - 1

Solution Because the second equation states that x = y - 1, we can substitute y - 1 for x in the first equation:

3x + 4y = 5

3(y - 1) + 4y = 5 Replace x with y - 1.

3y - 3 + 4y = 5 Simplify.

7y - 3 = 5

7y = 8

y = ? 8 7

?

Now use the value y = ? 8 7? in one of the original equations to find x. The simplest one to use is x = y - 1:

x = ? 8 7

? - 1

x = ? 1 7

?

Check that (? 1 7?, ? 8 7?) satisfies both equations. The solution set to the system is {(? 1 7?, ? 8 7?)}.

U Calculator Close-Up V

To check Example 1, graph

y1 = (5 - 3x)/4 and

y2 = x + 1.

Use the intersect feature of your calculator to find the point of intersection.

10

10 10

10 Now do Exercises 1–8

For substitution we must have one of the equations solved for x or y in terms of the other variable. In Example 1 we were given x = y - 1. So we replaced x with y - 1. In Example 2 we must rewrite one of the equations before substituting. Note how the five steps in the following strategy are used in Example 2.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 468

468 Chapter 7 Systems of Linear Equations 7-12

Strategy for Solving a System by Substitution

1. If necessary, solve one of the equations for one variable in terms of the other. Choose the equation that is easiest to solve for x or y.

2. Substitute into the other equation to eliminate one of the variables.

3. Solve the resulting equation in one variable.

4. Insert the solution found in the last step into one of the original equations and solve for the other variable.

5. Check your solution in both equations.

E X A M P L E 2 Solving a system by substitution Solve:

2x - 3y = 9

y - 4x = -8

Solution (1) Solve the second equation for y :

y - 4x = -8

y = 4x - 8

(2) Substitute 4x - 8 for y in the first equation:

2x - 3y = 9

2x - 3(4x - 8) = 9 Replace y with 4x - 8.

(3) Solve the equation for x:

2x - 12x + 24 = 9 Simplify.

-10x + 24 = 9

-10x = -15

x = ? -

-

1

1

5

0 ?

= ? 3 2

?

(4) Use the value x = ? 3 2

? in y = 4x - 8 to find y :

y = 4 · ? 3

2 ? - 8

= -2

(5) Check x = ? 3 2

? and y = -2 in both of the original equations:

2(? 3 2 ?) - 3(-2) = 9 Correct -2 - 4(? 3 2 ?) = -8 Correct

Since both are correct, the solution set to the system is {(? 3 2 ?, -2)}. Now do Exercises 9–16

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 469

7-13 7.2 The Substitution Method 469

U2V Dependent and Inconsistent Systems Examples 3 and 4 illustrate how to solve dependent and inconsistent systems by substitution.

E X A M P L E 4 A system with no solution Solve by substitution:

3x - 6y = 9

x = 2y + 5

Solution Use x = 2y + 5 to replace x in the first equation:

3x - 6y = 9

3(2y + 5) - 6y = 9 Replace x by 2y + 5.

6y + 15 - 6y = 9 Simplify.

15 = 9

No values for x and y will make 15 equal to 9. So there is no ordered pair that satisfies both equations. This system is inconsistent. It has no solution. The equations are the equations of parallel lines.

U Calculator Close-Up V

To check Example 4, graph y1 = (3x - 9)/6 and y2 = (x - 5)/2. Since the lines appear to be parallel, there is no solution to the system.

10

10 10

E X A M P L E 3 A system with infinitely many solutions Solve:

2(y - x) = x + y - 1

y = 3x - 1

Solution Because the second equation is solved for y, we will eliminate the variable y in the substi­ tution. Substitute y = 3x - 1 into the first equation:

2(3x - 1 - x) = x + (3x - 1) - 1

2(2x - 1) = 4x - 2

4x - 2 = 4x - 2

Every real number satisfies 4x - 2 = 4x - 2 because both sides are identical. So every real number can be used for x in the original system as long as we choose y = 3x - 1. The system is dependent. The graphs of these two equations are the same line. So the solution to the system is the set of all points on that line, {(x, y) I y = 3x - 1}.

Now do Exercises 17–20

10 Now do Exercises 21–26

U Helpful Hint V When solving a system by substitution we can recognize a dependent system or The purpose of Examples 3 and 4 an inconsistent system as follows. is to show what happens when substitution is used on dependent and inconsistent systems. If we had first Recognizing Dependent or Inconsistent Systems written the equations in slope- intercept form, we would see that the Substitution in a dependent system results in an equation that is always true. lines in Example 3 are the same and Substitution in an inconsistent system results in a false equation. the lines in Example 4 are parallel.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 470

470 Chapter 7 Systems of Linear Equations 7-14

U3V Applications Many of the problems that we solved in previous chapters had two unknown quanti­ ties, but we wrote only one equation to solve the problem. For problems with two unknown quantities we can use two variables and a system of equations.

E X A M P L E 5 Two investments Mrs. Robinson invested a total of $25,000 in two investments, one paying 6% and the other paying 8%. If her total income from these investments was $1790, then how much money did she invest in each?

Solution Let x represent the amount invested at 6%, and let y represent the amount invested at 8%. The following table organizes the given information.

Interest Amount Amount Rate Invested of Interest

First investment 6% x 0.06x Second investment 8% y 0.08y

Write one equation describing the total of the investments, and the other equation describ­ ing the total interest:

x + y = 25,000 Total investments

0.06x + 0.08y = 1790 Total interest

To solve the system, we solve the first equation for y:

y = 25,000 - x

Substitute 25,000 - x for y in the second equation:

0.06x + 0.08(25,000 - x) = 1790

0.06x + 2000 - 0.08x = 1790

-0.02x + 2000 = 1790

-0.02x = -210

x = ? -

-

0 2 . 1 0 0 2

?

= 10,500

Let x = 10,500 in the equation y = 25,000 - x to find y :

y = 25,000 - 10,500

= 14,500

Check these values for x and y in the original problem. Mrs. Robinson invested $10,500 at 6% and $14,500 at 8%.

U Helpful Hint V

In Chapter 2, we would have done Example 5 with one variable by letting x represent the amount invested at 6% and 25,000 - x represent the amount invested at 8%.

U Calculator Close-Up V

You can use a calculator to check the answers in Example 5:

Now do Exercises 55–84

dug84356_ch07a.qxd 9/18/10 8:55 AM Page 471

7-15 7.2 The Substitution Method 471

7 .2

Warm-Ups ▼

Fill in the blank. 1. The disadvantage of solving a system by is

inaccuracy.

2. In the method we eliminate a variable by substituting one equation into the other.

3. If substitution in a linear system results in a equation, then the system has exactly one solution.

4. If substitution results in an identity, then the system is .

5. If substitution results in an equation, then the system has no solution.

True or false? 6. Substituting y � 2x into x � 3y � 11 yields

x � 6x � 11. 7. A system of equations that has at least one solution is

consistent. 8. A consistent system with infinitely many solutions is

dependent. 9. An inconsistent system has no solutions.

10. No ordered pair satisfies y � 3x � 5 and y � 2x � 5.

Exercises

U Study Tips V • Students who have difficulty with a subject often schedule a class that meets one day per week so that they do not have to see it too

often. It is better to be in a class that meets more often for shorter time periods. • Students who explain things to others often learn from it. If you must work on math alone, try explaining things to yourself.

U1V Solving a System by Substitution

Solve each system by substitution. See Examples 1 and 2. See the Strategy for Solving a System by Substitution box on page 468.

1. y � x � 2 2. y � x � 4 x � y � 8 x � y � 12

3. x � y � 3 4. x � y � 1 x � y � 11 x � y � 7

5. y � x � 3 6. y � x � 5 2x � 3y � �11 x � 2y � 8

7. x � 2y � 4 8. x � y � 2 2x � y � 7 �2x � y � �1

9. 2x � y � 5 10. 5y � x � 0

5x � 2y � 8 6x � y � 29

11. x � y � 0 12. x � y � 6 3x � 2y � �5 3x � 4y � �3

13. x � y � 1 14. x � y � 2

4x � 8y � �4 3x � 6y � 8

15. 2x � 3y � 2 16. x � 2y � 1

4x � 9y � �1 3x � 10y � �1

U2V Dependent and Inconsistent Systems

Solve each system by substitution. Indicate whether each system is independent, dependent, or inconsistent. See Examples 1–4.

17. 21x � 35 � 7y 3x � y � 5

18. 2x � y � 3x 3x � y � 2y

19. x � 2y � �2

x � 2y � 8

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 472

7-16 472 Chapter 7 Systems of Linear Equations

20. y = -3x + 1

y = 2x + 4

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

22. y - 3 = 2(x - 1) y = 2x + 3

23. y + 1 = 5(x + 1) y = 5x - 1

24. 3x - 2y = 7

3x + 2y = 7

25. 2x + 5y = 5

3x - 5y = 6

26. x + 5y = 4 x + 5y = 4y

Solve each system by the graphing method shown in Section 7.1, and by substitution.

27. x + y = 5 28. x + y = 6 x - y = 1 2x - y = 3

29. y = x - 2 30. y = 2x - 3 y = 4 - x y = -x + 3

31. y = 3x - 2 32. x + y = 5 y - 3x = 1 y = 2 - x

Determine whether each system is independent, dependent, or inconsistent.

33. y = -4x + 3 34. y = -3x - 6 y = -4x - 6 y = 3x - 6

35. y = x 36. y = x x = y y = x + 5

37. y = x 38. y = 3x y = -x 3x - y = 0

39. x - y = 4 40. y = 1 x - y = 5 y + 3 = 4

Solve each system by the substitution method. 5

41. y = ?? x 42. 6x - 3y = 3 2

x + 3y = 3 10x = y + 7

43. x + y = 4 44. 3x - 6y = 5 x - y = 5 2y = 4x - 6

45. 2x - 4y = 0 46. -3x + 10y = 4 6x + 8y = 5 6x - 5y = 1

47. 3x + y = 2 48. x + 3y = 2 -x - 3y = 6 -x + y = 1

49. -9x + 6y = 3 50. x + 6y = -2 18x + 30y = 1 5x - 20y = 5

51. y = -2x 52. y = 2x 3y - x = 1 15x - 10y = -2

53. x = -6y + 1 54. x = -3y + 2 2y = -5x 7y = 3x

U3V Applications Write a system of two equations in two unknowns for each problem. Solve each system by substitution. See Example 5.

55. Rectangular patio. The length of a rectangular patio is twice the width. If the perimeter is 84 feet, then what are the length and width?

56. Rectangular lot. The width of a rectangular lot is 50 feet less than the length. If the perimeter is 900 feet, then what are the length and width?

57. Investing in the future. Mrs. Miller invested $20,000 and received a total of $1600 in interest. If she invested part of the money at 10% and the remainder at 5%, then how much did she invest at each rate?

58. Stocks and bonds. Mr. Walker invested $30,000 in stocks and bonds and had a total return of $2880 in one year. If his stock investment returned 10% and his bond invest­ ment returned 9%, then how much did he invest in each?

59. Gross receipts. Two of the highest grossing movies of all time were Titanic and Star Wars with total receipts of

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 473

7-17 7.2 The Substitution Method 473

$1062 million (www.movieweb.com). If the gross receipts supply is equal to the demand, the equilibrium price? See for Titanic exceeded the gross receipts for Star Wars by the accompanying figure. $140 million, then what were the gross receipts for each movie?

60. Tennis court dimensions. The singles court in tennis is four yards longer than it is wide. If its perimeter is 44 yards, then what are the length and width?

61. Mowing and shoveling. When Mr. Wilson came back from his vacation, he paid Frank $50 for mowing his lawn three times and shoveling his sidewalk two times. During Mr. Wilson’s vacation last year, Frank earned $45 for

Q ua

nt it

y (p

ou nd

s/ da

y)

1000

800

600

400

1 2 3 4 5 6

Point of equilibrium

Demand y 150x 900

Supply y 200x 60

200mowing the lawn two times and shoveling the sidewalk three times. How much does Frank make for mowing the lawn once? How much does Frank make for shoveling the sidewalk once?

62. Burgers and fries. Donna ordered four burgers and one order of fries at the Hamburger Palace. However, the waiter put three burgers and two orders of fries in the bag and charged Donna the correct price for three burgers and two orders of fries, $3.15. When Donna discovered the mistake, she went back to complain. She found out that the price for four burgers and one order of fries is $3.45 and decided to keep what she had. What is the price of one burger, and what is the price of one order of fries?

63. Racing rules. According to NASCAR rules, no more than 52% of a car’s total weight can be on any pair of tires. For optimal performance a driver of a 1150-pound car wants to have 50% of its weight on the left rear and left front tires and 48% of its weight on the left rear and right front tires. If the right front weight is determined to be 264 pounds, then what amount of weight should be on the left rear and left front? Are the NASCAR rules satisfied with this weight distribution?

64. Weight distribution. A driver of a 1200-pound car wants to have 50% of the car’s weight on the left front and left rear tires, 48% on the left rear and right front tires, and 51% on the left rear and right rear tires. How much weight should be on each of these tires?

65. Price of hamburger. A grocer will supply y pounds of ground beef per day when the retail price is x dollars per pound, where y = 200x + 60. Consumer studies show that consumer demand for ground beef is y pounds per day, where y = -150x + 900. What is the price at which the

0 Price of ground beef

(dollars/pound)

Figure for Exercise 65

66. Tweedle Dum and Dee. Tweedle Dum said to Tweedle Dee, “The sum of my weight and twice yours is 361 pounds.” Tweedle Dee said to Tweedle Dum, “Contrariwise the sum of my weight and twice yours is 362 pounds.” Find the weight of each.

67. Flying to Vegas. Two hundred people were on a charter flight to Las Vegas. Some paid $200 for their tickets and some paid $250. If the total revenue for the flight was $44,000, then how many tickets of each type were sold?

68. Annual concert. A total of 150 tickets were sold for the annual concert to students and nonstudents. Student tickets were $5 and nonstudent tickets were $8. If the total revenue for the concert was $930, then how many tickets of each type were sold?

69. Annual play. There were twice as many tickets sold to non- students than to students for the annual play. Student tickets were $6 and nonstudent tickets were $11. If the total revenue for the play was $1540, then how many tickets of each type were sold?

70. Soccer game. There were 1000 more students at the soccer game than nonstudents. Student tickets were $8.50 and nonstudent tickets were $13.25. If the total revenue for the game was $75,925, then how many tickets of each type were sold?

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 474

474 Chapter 7 Systems of Linear Equations 7-18

71. Mixing investments. Helen invested $40,000 and received a total of $2300 in interest after one year. If part of the money returned 5% and the remainder 8%, then how much did she invest at each rate?

72. Investing her bonus. Donna invested her $33,000 bonus and received a total of $970 in interest after one year. If part of the money returned 4% and the remainder 2.25%, then how much did she invest at each rate?

73. Mixing acid. A chemist wants to mix a 5% acid solution with a 25% acid solution to obtain 50 liters of a 20% acid solution. How many liters of each solution should be used?

74. Mixing fertilizer. A farmer wants to mix a liquid fertilizer that contains 2% nitrogen with one that contains 10% nitrogen to obtain 40 gallons of a fertilizer that contains 8% nitrogen. How many gallons of each fertilizer should be used?

75. Different interest rates. Mrs. Brighton invested $30,000 and received a total of $2300 in interest. If she invested part of the money at 10% and the remainder at 5%, then how much did she invest at each rate?

76. Different growth rates. The combined population of Marysville and Springfield was 25,000 in 2000. By 2005 the population of Marysville had increased by 10%, while Springfield had increased by 9%. If the total population increased by 2380 people, then what was the population of each city in 2000?

77. Toasters and vacations. During one week a land developer gave away Florida vacation coupons or toasters to 100 potential customers who listened to a sales presentation. It costs the developer $6 for a toaster and $24 for a Florida vacation coupon. If his bill for prizes that week was $708, then how many of each prize did he give away?

78. Ticket sales. Tickets for a concert were sold to adults for $3 and to students for $2. If the total receipts were $824 and twice as many adult tickets as student tickets were sold, then how many of each were sold?

79. Corporate taxes. According to Bruce Harrell, CPA, the amount of federal income tax for a class C corporation is deductible on the Louisiana state tax return, and the amount of state income tax for a class C corporation is

deductible on the federal tax return. So for a state tax rate of 5% and a federal tax rate of 30%, we have

state tax = 0.05(taxable income - federal tax)

and

federal tax = 0.30(taxable income - state tax).

Find the amounts of state and federal income taxes for a class C corporation that has a taxable income of $100,000.

80. More taxes. Use the information given in Exercise 79 to find the amounts of state and federal income taxes for a class C corporation that has a taxable income of $300,000. Use a state tax rate of 6% and a federal tax rate of 40%.

81. Cost accounting. The problems presented in this exercise and Exercise 82 are encountered in cost accounting. A company has agreed to distribute 20% of its net income N to its employees as a bonus; B = 0.20N. If the company has an income of $120,000 before the bonus, the bonus B is deducted from the $120,000 as an expense to determine net income; N = 120,000 - B. Solve the system of two equations in N and B to find the amount of the bonus.

82. Bonus and taxes. A company has an income of $100,000 before paying taxes and a bonus. The bonus B is to be 20% of the income after deducting income taxes T but before deducting the bonus. So,

B = 0.20(100,000 - T ).

Because the bonus is a deductible expense, the amount of income tax T at a 40% rate is 40% of the income after deducting the bonus. So,

T = 0.40(100,000 - B).

a) Use the accompanying graph to estimate the values of T and B that satisfy both equations.

b) Solve the system algebraically to find the bonus and the amount of tax.

83. Textbook case. The accompanying graph shows the cost of producing textbooks and the revenue from the sale of those textbooks.

a) What is the cost of producing 10,000 textbooks? b) What is the revenue when 10,000 textbooks are sold? c) For what number of textbooks is the cost equal to the

revenue? d) The cost of producing zero textbooks is called the fixed

cost. Find the fixed cost.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 475

7-19 7.2 The Substitution Method 475

Getting More Involved

100

T 0.40(100,000 B)

B 0.20(100,000 T)

85. Discussion100

Which of the following equations is not equivalent to B

on us

( in

th ou

sa nd

s of

d ol

la rs

)

80 2x - 3y = 6?

60 2 b) y = ?a) 3y - 2x = 6

40 ? x - 2 3

20

3 c) x = ?? y + 3 d) 2(x - 5) = 3y - 4

2

Figure for Exercise 83

x

A m

ou nt

(i n

m ill

io ns

o f

do lla

rs )

Number of textbooks (in thousands)

0

0.2 0.4 0.6 0.8 1.0 1.2

10 20 30 40

y

R 30x

C 10x 400,000

0 0 20 40 60 80 86. Discussion Taxes (in thousands of dollars)

Which of the following equations is inconsistent with the equation 3x + 4y = 8?

Figure for Exercise 82 3 a) y = ?? x + 2

4 b) 6x + 8y = 16

3 c) y = -?? x + 8

4 d) 3x - 4y = 8

Graphing Calculator Exercise

87. Life expectancy. Since 1950, the life expectancy of a U.S. male born in year x is modeled by the formula

y = 0.165x - 256.7,

84. Free market. The equations S = 5000 + 200x and D = 9500 - 100x express the supply S and the demand D, respectively, for a popular compact disc brand in terms of its price x (in dollars).

a) Graph the equations on the same coordinate system. b) What happens to the supply as the price increases? c) What happens to the demand as the price increases? d) The price at which supply and demand are equal is

called the equilibrium price. What is the equilibrium price?

and the life expectancy of a U.S. female born in year x is modeled by

y = 0.186x - 290.6

(National Center for Health Statistics, www.cdc.gov).

a) Find the life expectancy of a U.S. male born in 1975 and a U.S. female born in 1975.

b) Graph both equations on your graphing calculator for 1950 < x < 2050.

c) Will U.S. males ever catch up with U.S. females in life expectancy?

d) Assuming that these equations were valid before 1950, solve the system to find the year of birth for which U.S. males and females had the same life expectancy.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 476

476 Chapter 7 Systems of Linear Equations 7-20

Electricity is the flow of electrons through a circuit. It is measured in volts, amps, and watts. Volts measure the force that causes the electricity or electrons to flow. Amps measure the amount of electric current. Watts measure the amount of work done by a certain amount of cur­ rent at a certain force or voltage. The basic relationship is watts = amps · volts or W = A · V.

A circuit breaker is used as a safety device in a circuit. If the amperage exceeds a certain level, the breaker trips and prevents damage to the system. For example, suppose that 8 strings of Christmas lights each containing 25 bulbs that are 7 watts each are all plugged into one 120-volt circuit containing a 15-amp breaker. Will the breaker trip? The total wattage

is 8 · 25 · 7 or 1400 watts. Use A = W/V to get A = 1400/ 120 � 11.7. So the lights will not blow a 15-amp fuse. See the accompanying figure.

While houses use standard single-phase electricity, electri­ cal power companies may supply power for large users to trans­ formers through three-phase lines. The power in a three-phase system is measured in volt-amps. The formula used here is volt- amps = �3� · A · V. For example, suppose a large shopping mall has a 1,000,000 volt-amp transformer and the power company provides 25,000 volts to the mall’s transformer. Will this power trip a 20-amp breaker? Because A = volt-amps/ (�3� · V), we have A = 1,000,000/(�3� · 25,000) � 23.1 amps. So the 20-amp breaker will blow.

Math at Work Circuit Breakers A

m ps

20

15

10

5

500 1000 20001500 Watts

120-Volt Circuit

0

A W 120

Mid-Chapter Quiz Sections 7.1 through 7.2 Chapter 7

Determine whether (1, -2) is in the solution set to each system.

1. x - y = 3 2. x + y = -1 2x + y = 0 3x - y = 8

3. 5x + 12y = -19 5x + 12y = 6

Solve by graphing.

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

5. x - y = 8 x + y = 0

6. y = x + 6 x - y = -6

Solve by substitution.

7. y = 3x - 5 8. x + y = 6 2x + 5y = 9 3x - 5y = 26

9. 5x - y = 8 35x - 6 = 7y

Determine whether each system is independent, dependent, or inconsistent.

1 10. y = ??x - 7 11. y = 5x - 12

2 1

y = ??x + 5 y = 3x + 7 2 3

12. y = ??x + 1 4

4y = 3x + 4

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 477

7-21 7.3 The Addition Method 477

In This Section

U1V The Addition Method

U2V Equations Involving Fractions or Decimals

U3V Applications

7.3 The Addition Method

In Section 7.2, you used substitution to eliminate a variable in a system of equations. In this section, we see another method for eliminating a variable in a system of equations.

U1V The Addition Method In the addition method we eliminate a variable by adding the equations.

E X A M P L E 1 An independent system solved by addition Solve the system by the addition method:

3x - 5y = -9 4x + 5y = 23

Solution The addition property of equality allows us to add the same number to each side of an equation. We can also use the addition property of equality to add the two left sides and add the two right sides:

3x - 5y = -9 4x + 5y = 23

7x = 14 Add.

x = 2

The y-term was eliminated when we added the equations because the coefficients of the y-terms were opposites. Now use x = 2 in one of the original equations to find y. It does not matter which original equation we use. In this example we will use both equations to see that we get the same y in either case.

3x - 5y = -9 4x + 5y = 23

3(2) - 5y = -9 Replace x by 2. 4(2) + 5y = 23

6 - 5y = -9 Solve for y. 8 + 5y = 23

-5y = -15 5y = 15

y = 3 y = 3

Because 3(2) - 5(3) = -9 and 4(2) + 5(3) = 23 are both true, (2, 3) satisfies both equa­ tions. The solution set is {(2, 3)}.

U Calculator Close-Up V

To check Example 1, graph

y1 = (-9 - 3x)/-5 and

y2 = (23 - 4x)/5. Use the intersect feature to find the point of intersection of the two lines.

10

10

10

10

Now do Exercises 1–8

Actually the addition method can be used to eliminate any variable whose coefficients are opposites. If neither variable has coefficients that are opposites, then we use the multiplication property of equality to change the coefficients of the variables, as shown in Examples 2 and 3.

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 478

478 Chapter 7 Systems of Linear Equations 7-22

E X A M P L E 2 Using multiplication and addition Solve the system by the addition method:

2x - 3y = -13

5x - 12y = -46

Solution First examine the system to find the simplest way to eliminate a variable. Note that 5 is not a multiple of 2, but 12 is a multiple of 3. So if we multiply both sides of the first equation by -4, the coefficients of y will be 12 and -12, and y will be eliminated by addition:

(-4)(2x - 3y) = (-4)(-13) Multiply each side by -4.

5x - 12y = -46

-8x + 12y = 52 5x - 12y = -46 Add.

-3x = 6

x = -2

Replace x by -2 in one of the original equations to find y :

2x - 3y = -13

2(-2) - 3y = -13

-4 - 3y = -13

-3y = -9

y = 3

Because 2(-2) - 3(3) = -13 and 5(-2) - 12(3) = -46 are both true, the solution set is {(-2, 3)}.

Now do Exercises 9–12

U Calculator Close-Up V

Check Example 2 by graphing

y1 = (-13 - 2x)/(-3)

and

y2 = (-46 - 5x)/(-12).

-10

-10

10

10

E X A M P L E 3 Multiplying both equations before adding Solve each system by the addition method.

a) -2x + 3y = 6 b) -2x + 3y = 0 3x - 5y = -11 3x - 5y = 0

Solution a) Examine the coefficients. Since 3 is not a multiple of 2 and 5 is not a multiple of 3, we

can’t eliminate a variable by multiplying only one equation. However, multiplying the first equation by 3 and the second by 2 will give us -6x and 6x:

3(-2x + 3y) = 3(6) Multiply each side by 3.

2(3x - 5y) = 2(-11) Multiply each side by 2.

-6x + 9y = 18 6x - 10y = -22 Add.

-y = -4

y = 4

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 479

7-23 7.3 The Addition Method 479

Note that we could have eliminated y by multiplying by 5 and 3. Now insert y = 4 into one of the original equations to find x :

-2x + 3(4) = 6 Let y = 4 in -2x + 3y = 6.

-2x + 12 = 6

-2x = -6

x = 3

Check that (3, 4) satisfies both equations. The solution set is {(3, 4)}. b) Multiplying the first equation by 3, the second by 2, and then adding will eliminate x

as it did in part (a):

-2x + 3y = 0 Multiply by 3 -6x + 9y = 0

3x - 5y = 0 Multiply by 2 6x - 10y = 0

-y = 0

y = 0

If y = 0 in -2x + 3y = 0, we get -2x = 0 or x = 0. So the solution set is {(0, 0)}. Note that the graphs of these two equations intersect at the origin.

Now do Exercises 13–18

The strategy for solving an independent system by addition follows.

Strategy for the Addition Method

1. Write both equations in the same form (usually Ax + By = C).

2. If necessary multiply one or both equations by the appropriate integer to obtain opposite coefficients on one of the variables.

3. Add the equations to get an equation in one variable.

4. Solve the equation in one variable.

5. Substitute the value obtained for one variable into one of the original equa­ tions to obtain the value of the other variable.

6. Check the two values in both of the original equations.

We can identify dependent and inconsistent systems in the same way that we did for the substitution method. If the result of the addition is an identity, the system is depen­ dent and there are infinitely many solutions. If the result of the addition is a false equa­ tion, the system is inconsistent and there are no solutions. When you use addition, make sure that the equations are in the same form with the variables and equal signs aligned.

E X A M P L E 4 Solving dependent and inconsistent systems by addition Solve each system by addition:

a) 2x - 3y = 9 b) -4y = 5x + 7 6y = 4x - 18 4y + 5x = 12

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 480

480 Chapter 7 Systems of Linear Equations 7-24

Solution a) First rewrite 6y = 4x - 18 as -4x + 6y = -18 so that it is in the same form as the

first equation:

2x - 3y = 9

-4x + 6y = -18

Now examine the coefficients. Multiply the first equation by 2 to get 4x - 6y = 18 and add:

4x - 6y = 18

-4x + 6y = -18

0 = 0

Because the result of the addition is an identity, the equations are dependent and there are infinitely many solutions. The solution set is {(x, y) | 2x - 3y = 9}.

b) Rewrite the first equation -4y = 5x + 7 as -4y - 5x = 7 to get the same form as the second. Now add:

-4y - 5x = 7

4y + 5x = 12

0 = 19

Because the result of the addition is a false equation, the system is inconsistent. There are no solutions to the system. The solution set is the empty set, 0.

U Calculator Close-Up V

To check Example 4(b), graph

y1 = (5x + 7)/-4

and

y2 = (-5x + 12)/4.

Since the lines appear to be parallel, the graph supports the conclusion that the system is inconsistent.

10

10

10

Now do Exercises 25–30 10

U2V Equations Involving Fractions or Decimals When a system of equations involves fractions or decimals, we can use the multi­ plication property of equality to eliminate the fractions or decimals.

E X A M P L E 5 A system with fractions Solve the system:

? 1 2

? x - ? 2

3 ? y = 7

? 2

3 ? x - ?

3

4 ? y = 11

Solution Since 2 and 3 both divide evenly into 6, multiplying the first equation by 6 will eliminate its fractions. Since 3 and 4 both divide evenly into 12, multiplying the second equation by 12 will eliminate its fractions:

6 (? 1 2 ? x - ? 2

3 ? y) = 6(7) → 3x - 4y = 42

12 (? 2 3 ? x - ? 3

4 ? y) = 12(11) → 8x - 9y = 132

U Calculator Close-Up V

To check Example 5, graph

y1 = (7 - (1/2)x)/(-2/3)

and

y2 = (11 - (2/3)x)/(-3/4).

The lines appear to intersect at (30, 12).

10

10

20

40

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 481

7-25 7.3 The Addition Method 481

Now examine the coefficients in the two new equations. Both equations will have to be mul­ tiplied to eliminate x or y. To eliminate x, multiply the first by -8 and the second by 3:

-8(3x - 4y) = -8(42) → -24x + 32y = -336 3(8x - 9y) = 3(132) → 24x - 27y = 396

5y = 60

y = 12

Substitute y = 12 into the first of the original equations:

? 1 2

? x - ? 2 3

? (12) = 7

? 1 2

? x - 8 = 7

? 1 2

? x = 15

x = 30

Check (30, 12) in the original system. The solution set is {(30, 12)}.

Now do Exercises 31–38

Note that there are many ways to proceed in Example 5. We multiplied first to eliminate the fractions and second to eliminate a variable. That is usually the simplest approach. However, if you multiply the first equation by -48 and the second by 36, you would only have to multiply once.

E X A M P L E 6 A system with decimals Solve the system:

0.05x + 0.7y = 40

x + 0.4y = 120

Solution Multiplying by 10 or 100 moves the decimal point one or two places to the right, respec­ tively. So multiplying the first equation by 100 and the second by 10 will eliminate all of the decimals:

100(0.05x + 0.7y) = 100(40) → 5x + 70y = 4000 10(x + 0.4y) = 10(120) → 10x + 4y = 1200

Examine the coefficients. Since 10 is a multiple of 5, we can eliminate x by multiplying the first equation by -2:

-2(5x + 70y) = -2(4000) → -10x - 140y = -8000 10x + 4y = 1200 → 10x + 4y = 1200

-136y = -6800

y = 50 Use y = 50 in x + 0.4y = 120 to find x:

x + 0.4(50) = 120

x + 20 = 120

x = 100

Check (100, 50) in the original system. The solution set is {(100, 50)}.

U Calculator Close-Up V

Check Example 6 by graphing

y1 = (40 - 0.05x)/(0.7)

and

y2 = (120 - x)/(0.4).

-20

-100

100

200

Now do Exercises 39–46

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 482

482 Chapter 7 Systems of Linear Equations 7-26

We have seen three methods for solving a system of two linear equations in two variables. For some systems the method you choose can make a difference. The following summary should help you decide which method to use.

Summary of the Methods

Graphing It is impossible to identify the solution from a graph unless it is very simple. Graphing helps us understand the difference between independent, dependent, and inconsistent systems. Graphing works well with a graphing calculator.

Substitution Substitution is used when one of the equations is solved for one of the variables or when it is easy to isolate one of the variables in an equation.

Addition Addition is used when both equations are in the same form and it is easy to eliminate a variable by multiplying and adding.

U3V Applications Any system of two linear equations in two variables can be solved by either the addition method or substitution. In applications we use whichever method appears to be the sim­ pler for the problem at hand.

Fajitas and burritos At the Cactus Cafe the total price for four fajita dinners and three burrito dinners is $48, and the total price for three fajita dinners and two burrito dinners is $34. What is the price of each type of dinner?

Solution Let x represent the price (in dollars) of a fajita dinner, and let y represent the price (in dollars) of a burrito dinner. We can write two equations to describe the given information:

4x + 3y = 48 3x + 2y = 34

Because 12 is the least common multiple of 4 and 3 (the coefficients of x), we multiply the first equation by -3 and the second by 4:

-3(4x + 3y) = -3(48) Multiply each side by -3. 4(3x + 2y) = 4(34) Multiply each side by 4.

-12x - 9y = -144 12x + 8y = 136 Add.

-y = -8 y = 8

To find x, use y = 8 in the first equation 4x + 3y = 48:

4x + 3(8) = 48 4x + 24 = 48

4x = 24 x = 6

So the fajita dinners are $6 each, and the burrito dinners are $8 each. Check this solution in the original problem.

E X A M P L E 7

U Helpful Hint V

You can see from Example 7 that the standard form Ax + By = C occurs naturally in accounting. This form will occur whenever we have the price of each item and a quantity of two items and want to express the total cost.

Now do Exercises 65–70

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 483

7-27 7.3 The Addition Method 483

Warm-Ups ▼

Fill in the blank. 1. In the method we eliminate a variable by

adding the equations.

2. If addition in a linear system results in a equation, then the system has exactly one solution.

3. If addition results in an identity, then the system is .

4. If addition results in an equation, then the system has no solution.

5. If addition results in a equation, then the two lines intersect at exactly one point.

6. If addition results in an , then the two lines have the same graph.

7. If addition results in an equation, then the two lines are parallel.

True or false? 8. To solve 3x - y = 9 and 3x + y = 6 by addition we

simply add the equations. 9. To solve 2x + 7y = 5 and 3x + 2y = 8 by addition we

multiply the first equation by 3, the second by 2, and then add.

10. Both (0, -10) and (5, 0) satisfy 4x - 2y = 20 and 4x + 2y = 20.

11. The system 4x - 5y = 9 and -4x + 5y = -9 has no solution.

12. Either addition or substitution could be used to solve 2x - y = 5 and 3x + 2y = 9

13. To eliminate fractions, multiply both sides of the equation by the least common denominator.

14. Either variable can be eliminated by the addition method.

E X A M P L E 8 Mixing cooking oil Canola oil is 7% saturated fat, and corn oil is 14% saturated fat. Crisco sells a blend, Crisco Canola and Corn Oil, which is 11% saturated fat. How many gallons of each type of oil must be mixed to get 280 gallons of this blend?

Solution Let x represent the number of gallons of canola oil, and let y represent the number of gallons of corn oil. Make a table to summarize all facts:

Since the total amount of oil is 280 gallons, we have x + y = 280. Since the total amount of fat is 30.8 gallons, we have 0.07x + 0.14y = 30.80. Since we can easily solve x + y = 280 for y, we choose substitution to solve the system. Substitute y = 280 - x into the second equation:

0.07x + 0.14(280 - x) = 30.80 Substitution

0.07x + 39.2 - 0.14x = 30.80 Distributive property

-0.07x = -8.4

x = ? -

-

0 8 . . 0 4 7

? = 120

If x = 120 and y = 280 - x, then y = 280 - 120 = 160. Check that

0.07(120) + 0.14(160) = 30.8.

So it takes 120 gallons of canola oil and 160 gallons of corn oil to make 280 gallons of Crisco Canola and Corn Oil.

Now do Exercises 71–78

Amount (gallons) % fat Amount of Fat (gallons)

Canola oil x 7 0.07x

Corn oil y 14 0.14y

Canola and Corn Oil 280 11 0.11(280) or 30.8

dug84356_ch07a.qxd 9/18/10 8:55 AM Page 484

7 .3 Exercises

U Study Tips V • Don’t expect to understand a topic the first time you see it. Learning mathematics takes time, patience, and repetition. • Keep reading the text, asking questions, and working problems. Someone once said, “All math is easy once you understand it.”

U1V The Addition Method

Solve each system by addition. See Examples 1–3. See the Strategy for the Addition Method box on page 479.

1. x - y = 1 2. x + y = 7 x + y = 7 x - y = 9

3. 3x - 4y = 11 4. 7x - 5y = -1 -3x + 2y = -7 -3x + 5y = 9

5. x - y = 12 6. x - 2y = -1 2x + y = 3 -x + 5y = 4

7. 3x - y = 5 8. -x + 2y = 4 5x + y = -2 x - 5y = 1

9. 2x - y = -5 10. 3x + 5y = -11 3x + 2y = 3 x - 2y = 11

11. -3x + 5y = 1 12. 7x - 4y = -3 9x - 3y = 5 x + 2y = 3

13. 2x - 5y = 13 14. 3x + 4y = -5 3x + 4y = -15 5x + 6y = -7

15. 2x = 3y + 11 16. 2x = 2 - y 7x - 4y = 6 3x + y = -1

17. x + y = 48 18. x + y = 13 12x + 14y = 628 22x + 36y = 356

Use a calculator to check whether the given ordered pair satis­ fies both equations of the given system.

19. (-45, 16) 3x + 2y = -103 5x - 8y = -353

20. (502, 388) -3x + 5y = 434

6x - 7y = 296

21. (42, 99)

2 5 3x + 33 y = 733 3 11

1 2 - 3 3x + 3 y = 93

3 9

22. (16.5, 25.6) 1 5

3x + 3 y = 69.533 3 2 4 3 3 x - 3 y = 63 3 5 4

23. (34.56, 59.66) 0.02x + 0.03y = 2.481

0.8x + 0.9y = 81.342

24. (40,000, 120,000) 0.08x + 0.12y = 17,600

x + y = 160,000

Solve each system by the addition method. Determine whether the equations are independent, dependent, or inconsistent. See Example 4.

25. 3x - 4y = 9 26. x - y = 3 -3x + 4y = 12 -6x + 6y = 17

27. 5x - y = 1 28. 4x + 3y = 2 10x - 2y = 2 -12x - 9y = -6

29. 2x - y = 5 30. -3x + 2y = 8 2x + y = 5 3x + 2y = 8

U2V Equations Involving Fractions or Decimals

Solve each system by the addition method. See Examples 5 and 6.

1 1 3 3x 2y

31. 3 x + 3 y = 5 32. 3 - 33 = 103 3 4 3 2 3

1 1 x - y = 6 3 x + 3 y = -13 3

2 2

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 485

7.3 The Addition Method 485

59. x - y = 0 60. 5x - 4y = 9 x + y = 2x 8y - 10x = -18

For each system find the value of a so that the solution set to the system is {(2, 3)}.

61. x + y = 5 62. 2x - y = 1 x - y = a ax + y = 13

For each system find the values of a and b so that the solution set to the system is {(5, 12)}.

63. y = ax + 2 64. y = 3x + a y = bx + 17 y = -2x + b

U3V Applications

Write a system of two equations in two unknowns for each problem. Solve each system by the method of your choice. See Examples 7 and 8.

65. Two numbers. The sum of two numbers is 12 and their difference is 2. Find the numbers.

66. Two more numbers. The sum of two numbers is 11 and their difference is 6. Find the numbers.

67. Paper size. The length of a rectangular piece of paper is 2.5 inches greater than the width. The perimeter is 39 inches. Find the length and width.

68. Photo size. The length of a rectangular photo is 2 inches greater than the width. The perimeter is 20 inches. Find the length and width.

69. Buy and sell. Cory buys and sells baseball cards on eBay. He always buys at the same price and then sells the cards for $2 more than he buys them. One month he broke even after buying 56 cards and selling 49. Find his buying price and selling price.

70. Jay Leno’s garage. Jay Leno’s collection of cars and motorcycles totals 187. When he checks the air in the tires, he has 588 tires to check. How many cars and how many motorcycles does he own? Assume that the cars all have four tires and the motorcycles have two.

71. Coffee and doughnuts. On Monday, Archie paid $3.40 for three doughnuts and two coffees. On Tuesday he paid $3.60 for two doughnuts and three coffees. On Wednesday he was

7-29

x y 33. ?? - ?? = -4

4 3 x y ?? + ?? = 0 8 6

1 1 35. ?? x + ?? y = 5

8 4 1 1

?? x + ?? y = 7 16 2

1 1 1 37. ?? x + ?? y = ??

3 2 3 5 3 1 ?? x - ?? y = ?? 6 4 6

39. 0.05x + 0.10y = 1.30 x + y = 19

41. x + y = 1200 0.12x + 0.09y = 120

43. 1.5x - 2y = -0.25 3x + 1.5y = 6.375

45. 0.24x + 0.6y = 0.58 0.8x - 0.12y = 0.52

Miscellaneous

34.

36.

38.

40.

42.

44.

x y 5 ?? - ?? = -?? 3 2 6 x y 3 ?? - ?? = -?? 5 3 5

3 5 ?? x + ?? y = 27 7 9 1 2 ?? x + ?? y = 7 9 7

2 5 1 ?? x + ?? y = ?? 3 6 4 1 1 1 ?? x - ?? y = -?? 5 10 10

0.1x + 0.06y = 9 0.09x + 0.5y = 52.7

x - y = 100 0.20x + 0.06y = 150

3x - 2.5y = 7.125 2.5x - 3y = 7.3125

46. 0.18x + 0.27y = 0.09 0.06x - 0.54y = -0.04

Solve each system by substitution or addition, whichever is easier.

47. y = x + 1 2x - 5y = -20

49. x - y = 19 2x + y = -13

51. 2y = x + 2 x = y - 1

53. 2y - 3x = -1 5y + 3x = 29

55. 6x + 3y = 4 2

y = ?? x 3

57. y = 3x + 1

1 x = ?? y + 5

3

48. y = 3x - 4 x + y = 32

50. x + y = 3 7x - y = 29

52. 2y - x = 3 x = 3y - 5

54. y - 5 = 2x y - 9 = -2x

56. 3x - 2y = 2 2

x = ?? y 9 2

58. y = -??x - 3 3 3

x = -?? y + 9 2

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 486

486 Chapter 7 Systems of Linear Equations 7-30

tired of paying the tab and went out for coffee by himself. b) Write a system of equations and solve it algebraically to What was his bill for one doughnut and one coffee? find the exact amount of each type that should be used to

obtain 50 pounds of double-dark-peanut fudge.

Figure for Exercise 71

1.5

2

0.5

1

0.5 1 1.5 2

3 doughnuts 2 coffees

$3.40

2 doughnuts 3 coffees

$3.60

C of

fe e

pr ic

e (i

n do

lla rs

)

Doughnut price (in dollars)

Pe an

ut b

ut te

r fu

dg e

(p ou

nd s)

60

0

20

40

0 20 40 3010 50 Double-dark fudge (pounds)

Total fudge

Total fat

72. Books and magazines. At Gwen’s garage sale, all books were one price, and all magazines were another price. Harriet bought four books and three magazines for $1.45, and June bought two books and five magazines for $1.25. What was the price of a book and what was the price of a magazine?

73. Boys and girls. One-half of the boys and one-third of the girls of Freemont High attended the homecoming game, whereas one-third of the boys and one-half of the girls attended the homecoming dance. If there were 570 students at the game and 580 at the dance, then how many students are there at Freemont High?

74. Girls and boys. There are 385 surfers in Surf City. Two- thirds of the boys are surfers and one-twelfth of the girls are surfers. If there are two girls for every boy, then how many boys and how many girls are there in Surf City?

75. Nickels and dimes. Winborne has 35 coins consisting of dimes and nickels. If the value of his coins is $3.30, then how many of each type does he have?

76. Pennies and nickels. Wendy has 52 coins consisting of nickels and pennies. If the value of the coins is $1.20, then how many of each type does she have?

77. Blending fudge. The Chocolate Factory in Vancouver blends its double-dark-chocolate fudge, which is 35% fat, with its peanut butter fudge, which is 25% fat, to obtain double-dark-peanut fudge, which is 29% fat.

a) Use the accompanying graph to estimate the number of pounds of each type that must be mixed to obtain 50 pounds of double-dark-peanut fudge.

Figure for Exercise 77

78. Low-fat yogurt. Ziggy’s Famous Yogurt blends regular yogurt that is 3% fat with its no-fat yogurt to obtain low- fat yogurt that is 1% fat. How many pounds of regular yogurt and how many pounds of no-fat yogurt should be mixed to obtain 60 pounds of low-fat yogurt?

79. Keystone state. Judy averaged 42 miles per hour (mph) driv­ ing from Allentown to Harrisburg and 51 mph driving from Harrisburg to Pittsburgh. See the accompanying figure. If she drove a total of 288 miles in 6 hours, then how long did it take her to drive from Harrisburg to Pittsburgh?

Figure for Exercise 79

Pittsburgh Harrisburg

42 mph51 mph Allentown

80. Empire state. Spike averaged 45 mph driving from Rochester to Syracuse and 49 mph driving from Syracuse to Albany. If he drove a total of 237 miles in 5 hours, then how far is it from Syracuse to Albany?

81. Probability of rain. The probability of rain tomorrow is four times the probability that it does not rain tomorrow. The probability that it rains plus the probability that it does not rain is 1. What is the probability that it rains tomorrow?

82. Super Bowl contender. The probability that San Francisco plays in the next Super Bowl is nine times the probability

dug84356_ch07a.qxd 9/14/10 1:54 PM Page 487

7-31

that they do not play in the next Super Bowl. The probability that San Francisco plays in the next Super Bowl plus the probability that they do not play is 1. What is the probability that San Francisco plays in the next Super Bowl?

83. Rectangular lot. The width of a rectangular lot is 75% of its length. If the perimeter is 700 meters, then what are the length and width?

84. Fence painting. Darren and Douglas must paint the 792-foot fence that encircles their family home. Because Darren is older, he has agreed to paint 20% more than Douglas. How much of the fence will each boy paint?

Getting More Involved

85. Discussion

Explain how you decide whether it is easier to solve a system by substitution or addition.

7.4 Systems of Linear Equations in Three Variables 487

86. Exploration

a) Write a linear equation in two variables that is satisfied by (-3, 5).

b) Write another linear equation in two variables that is satisfied by (-3, 5).

c) Are your equations independent or dependent? d) Explain how to select the second equation so that it will

be independent of the first.

87. Exploration

a) Make up a system of two linear equations in two variables such that both (-1, 2) and (4, 5) are in the solution set.

b) Are your equations independent or dependent? c) Is it possible to find an independent system that is satis­

fied by both ordered pairs? Explain.

7.4 Systems of Linear Equations in Three Variables

The techniques that you learned in Sections 7.2 and 7.3 can be extended to systems of equations in more than two variables. In this section, we use elimination of variables to solve systems of equations in three variables.

In This Section

U1V Definition

U2V Solving a System by Elimination

U3V Dependent and Inconsistent Systems

U4V Applications U 1V Definition

The equation 5x - 4y = 7 is called a linear equation in two variables because its graph is a straight line. The equation 2x + 3y - 4z = 12 is similar in form, and so it is a linear equation in three variables. An equation in three variables is graphed in a three-dimensional coordinate system. The graph of a linear equation in three variables is a plane, not a line. We will not graph equations in three variables in this text, but we can solve systems without graphing. In general, we make the following definition.

Linear Equation in Three Variables

If A, B, C, and D are real numbers, with A, B, and C not all zero, then

Ax + By + Cz = D

is called a linear equation in three variables.

U2V Solving a System by Elimination A solution to an equation in three variables is an ordered triple such as (-2, 1, 5), where the first coordinate is the value of x, the second coordinate is the value of y, and the third coordinate is the value of z. There are infinitely many solutions to a linear equation in three variables.

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 488

488 Chapter 7 Systems of Linear Equations 7-32

The solution to a system of equations in three variables is the set of all ordered triples that satisfy all of the equations of the system. The techniques for solving a system of linear equations in three variables are similar to those used on systems of linear equations in two variables. We eliminate variables by either substitution or addition.

A linear system with a single solution Solve the system:

(1) x + y - z = -1

(2) 2x - 2y + 3z = 8

(3) 2x - y + 2z = 9

Solution We can eliminate y from Eqs. (1) and (2) by multiplying Eq. (1) by 2 and adding it to Eq. (2):

2x + 2y - 2z = -2 Eq. (1) multiplied by 2

2x - 2y + 3z = 8 Eq. (2)

(4) 4x + z = 6

Now we must eliminate the same variable, y, from another pair of equations. Eliminate y from Eqs. (1) and (3) by simply adding them:

x + y - z = -1 Eq. (1)

2x - y + 2z = 9 Eq. (3)

(5) 3x + z = 8

Equations (4) and (5) give us a system with two variables. We now solve this system. Eliminate z by multiplying Eq. (4) by -1 and adding the equations:

-4x - z = -6 Eq. (4) multiplied by -1

3x + z = 8 Eq. (5)

-x = 2

x = -2

Now that we have x, we can replace x by -2 in Eq. (5) to find z:

3x + z = 8 Eq. (5) 3(-2) + z = 8

-6 + z = 8 z = 14

Now replace x by -2 and z by 14 in Eq. (1) to find y:

x + y - z = -1 Eq. (1) -2 + y - 14 = -1 x = -2, z = 14

y - 16 = -1 y = 15

Check that (-2, 15, 14) satisfies all three of the original equations. The solution set is {(-2, 15, 14)}.

U Calculator Close-Up V

You can use a calculator to check that (-2, 15, 14) satisfies all three equa­ tions of the original system.

E X A M P L E 1

Now do Exercises 1–4

Note that we could have eliminated any one of the three variables in Example 1 to get a system of two equations in two variables. We chose to eliminate y first because it was the easiest to eliminate. The strategy that we follow for solving a system of three linear equations in three variables is stated as follows:

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 489

7-33 7.4 Systems of Linear Equations in Three Variables 489

Strategy for Solving a System in Three Variables

1. Use substitution or addition to eliminate any one of the variables from a pair of equations of the system. Look for the easiest variable to eliminate.

2. Eliminate the same variable from another pair of equations of the system.

3. Solve the resulting system of two equations in two unknowns.

4. After you have found the values of two of the variables, substitute into one of the original equations to find the value of the third variable.

5. Check the three values in all of the original equations.

In Example 2, we use a combination of addition and substitution.

E X A M P L E 2

U Helpful Hint V

In Example 2 we chose to eliminate y first. Try solving Example 2 by first eliminating z. Write z = 2 - 2y, and then substitute 2 - 2y for z in Eqs. (1) and (2).

Using addition and substitution Solve the system: (1) x + y = 4

(2) 2x - 3z = 14 (3) 2y + z = 2

Solution From Eq. (1) we get y = 4 - x. If we substitute y = 4 - x into Eq. (3), then Eqs. (2) and (3) will be equations involving x and z only.

(3) 2y + z = 2 2(4 - x) + z = 2 Replace y by 4 - x.

8 - 2x + z = 2 Simplify. (4) -2x + z = -6

Now solve the system consisting of Eqs. (2) and (4) by addition:

2x - 3z = 14 Eq. (2) -2x + z = -6 Eq. (4)

-2z = 8 z = -4

Use Eq. (3) to find y:

2y + z = 2 Eq. (3) 2y + (-4) = 2 Let z = -4.

2y = 6

y = 3

Use Eq. (1) to find x:

x + y = 4 Eq. (1) x + 3 = 4 Let y = 3.

x = 1

Check that (1, 3, -4) satisfies all three of the original equations. The solution set is {(1, 3, -4)}.

Now do Exercises 5–20

U Calculator Close-Up V

Check that (1, 3, -4) satisfies all three equations in Example 2.

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 490

490 Chapter 7 Systems of Linear Equations 7-34

CAUTION In solving a system in three variables it is essential to keep your work organized and neat. Writing short notes that explain your steps (as was done in the examples) will allow you to go back and check your work.

U3V Dependent and Inconsistent Systems The graph of any equation in three variables can be drawn on a three-dimensional coordinate system. The graph of a linear equation in three variables is a plane. To solve a system of three linear equations in three variables by graphing, we would have to draw the three planes and then identify the points that lie on all three of them. This method would be difficult even when the points have simple coordinates. So we will not attempt to solve these systems by graphing.

In Section 7.1 we classified systems of two linear equations in two variables as consistent if the system had at least one solution and inconsistent if the system had no solutions. A consistent system with exactly one solution is independent, and a consis­ tent system with infinitely many solutions is dependent. We use the same terminology with systems of three linear equations in three variables. The only difference here is that there are more possibilities for the graphs of the dependent and inconsistent systems. Even though we don’t solve systems in three variables by graphing, the graphs in Fig. 7.7 will help you to better understand these systems.

(a) (b) (c) (d)

Figure 7.7

In most of the problems that we will solve, the planes intersect at a single point, as in Fig. 7.7(a). The solution set contains exactly one ordered triple, and the system is independent.

If the intersection of the three planes is a line or a plane, then the solution set is infi­ nite and the system is dependent. There are three possibilities. All three planes could intersect along a line as shown in Fig. 7.7(b). All three planes could be the same. In which case, all points on that plane satisfy the system. We could also have two equa­ tions for the same plane with the third plane intersecting it along a line.

If there are no points in common to all three planes, then the system is inconsis­ tent. The system will be inconsistent if at least two of the planes are parallel as shown in Fig. 7.7(c) and (d). There is one other configuration for an inconsistent system that is not shown here. See if you can find it.

We will not solve systems corresponding to all of the possible configurations described for the planes. Examples 3 and 4 illustrate two of these cases.

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 491

7-35 7.4 Systems of Linear Equations in Three Variables 491

E X A M P L E 4 A system with no solutions Solve the system:

(1) x + y - z = 5

(2) 3x - 2y + z = 8

(3) 2x + 2y - 2z = 7

Solution We can eliminate the variable z from Eqs. (1) and (2) by adding them:

x + y - z = 5 Eq. (1)

3x - 2y + z = 8 Eq. (2)

4x - y = 13

To eliminate z from Eqs. (1) and (3), multiply Eq. (1) by -2 and add the resulting equa­ tion to Eq. (3):

-2x - 2y + 2z = -10 Eq. (1) multiplied by -2

2x + 2y - 2z = 7 Eq. (3)

0 = -3

Because the last equation is false, the system is inconsistent. The solution set is the empty set.

Now do Exercises 23–34

U Helpful Hint V

If you recognize that multiplying Eq. (1) by -3 will produce Eq. (2), and multiplying Eq. (1) by 2 will produce Eq. (3), then you can conclude that all three equations are equivalent and there is no need to add the equations.

E X A M P L E 3 A system with infinitely many solutions Solve the system:

(1) 2x - 3y - z = 4

(2) -6x + 9y + 3z = -12

(3) 4x - 6y - 2z = 8

Solution We will first eliminate x from Eqs. (1) and (2). Multiply Eq. (1) by 3 and add the resulting equation to Eq. (2):

6x - 9y - 3z = 12 Eq. (1) multiplied by 3

-6x + 9y + 3z = -12 Eq. (2)

0 = 0

The last statement is an identity. The identity occurred because Eq. (2) is a multiple of Eq. (1). In fact, Eq. (3) is also a multiple of Eq. (1). These equations are dependent. They are all equations for the same plane. The solution set is the set of all points on that plane,

{(x, y, z) � 2x - 3y - z = 4}. Now do Exercises 21–22

U4V Applications Problems involving three unknown quantities can often be solved by using a system of three equations in three variables.

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 492

492 Chapter 7 Systems of Linear Equations 7-36

E X A M P L E 5 Finding three unknown rents Theresa took in a total of $1240 last week from the rental of three condominiums. She had to pay 10% of the rent from the one-bedroom condo for repairs, 20% of the rent from the two-bedroom condo for repairs, and 30% of the rent from the three-bedroom condo for repairs. If the three-bedroom condo rents for twice as much as the one-bedroom condo and her total repair bill was $276, then what is the rent for each condo?

Solution Let x, y, and z represent the rent on the one-bedroom, two-bedroom, and three-bedroom condos, respectively. We can write one equation for the total rent, another equation for the total repairs, and a third equation expressing the fact that the rent for the three-bedroom condo is twice that for the one-bedroom condo:

x + y + z = 1240

0.1x + 0.2y + 0.3z = 276

z = 2x

Substitute z = 2x into both of the other equations to eliminate z:

x + y + 2x = 1240 0.1x + 0.2y + 0.3(2x) = 276

3x + y = 1240 0.7x + 0.2y = 276

-2(3x + y) = -2(1240) Multiply each side by -2. 10(0.7x + 0.2y) = 10(276) Multiply each side by 10.

-6x - 2y = -2480 7x + 2y = 2760 Add.

x = 280

z = 2(280) = 560 Because z = 2x

280 + y + 560 = 1240 Because x + y + z = 1240

y = 400

Check that (280, 400, 560) satisfies all three of the original equations. The condos rent for $280, $400, and $560 per week.

U Helpful Hint V

A problem involving two unknowns can often be solved with one variable as in Chapter 2. Likewise, you can often solve a problem with three unknowns using only two variables. Solve Example 5 by letting a, b, and 2a be the rent for a one-bedroom, two-bedroom, and a three-bedroom condo.

U Calculator Close-Up V

Check that (280, 400, 560) satisfies all three equations in Example 5.

Now do Exercises 51–64

Warm-Ups ▼

Fill in the blank. 1. An equation of the form Ax + By + Cz = D where A,

B, and C are not all zero, is a equation in three variables.

2. The triple (a, b, c) corresponds to a point in a three-dimensional coordinate system.

3. A to a linear system in three variables is an ordered triple that satisfies all of the equations.

4. To solve a linear system in three variables use or to eliminate variables.

5. The graph of a linear equation in three variables is a in a three-dimensional coordinate system.

dug84356_ch07b.qxd 9/20/10 12:18 PM Page 493

7-37 7.4 Systems of Linear Equations in Three Variables 493

7 .4

6. For an system of three linear equations in three variables the planes intersect at a single point.

True or false? 7. The point (1, �2, 3) satisfies x � y � z � 4. 8. The point (4, 1, 1) is the only solution to x � y � z � 4.

9. The point (1, �1, 2) satisfies x � y � z � 2 and 2x � y � z � �1.

U Study Tips V

10. Two distinct planes are either parallel or intersect at a single point.

11. The equations 3x � 2y � 6z � 4 and �6x � 4y � 12z � �8 are dependent.

12. The graph of y � 2x � 3z � 4 is a line. 13. The value of x nickels, y dimes, and z quarters is

0.05x � 0.10y � 0.25z cents. 14. If x � �2, z � 3, and x � y � z � 6, then y � 7.

Exercises

• Finding out what happened in class and attending class are not the same. Attend every class and be attentive. • Don’t just take notes and let your mind wander. Use class time as a learning time.

U2V Solving a System by Elimination

Solve each system of equations. See Examples 1 and 2. See the Strategy for Solving a System in Three Variables box on page 489.

1. x � y � z � 9 2. x � y � z � 4 y � z � 7 y � 6

z � 4 y � z � 13

3. x � y � z � 10 4. x � y � z � 6 x � y � �1 y � z � 11 x � y � 5 y � z � 3

5. x � y � z � 6 6. x � y � z � 0 x � y � z � 2 x � y � z � 2 x � y � z � �4 x � y � z � 0

7. x � y � z � 2 8. 2x � y � 3z � 14 x � 2y � z � 6 x � y � 2z � �5

2x � y � z � 5 3x � y � z � 2

9. x � 2y � 4z � 3 10. 2x � 3y � z � 13 x � 3y � 2z � 6 �3x � 2y � z � �4 x � 4y � 3z � �5 4x � 4y � z � 5

11. 2x � y � z � 10 12. x � 3y � 2z � �11 3x � 2y � 2z � 7 2x � 4y � 3z � �15

x � 3y � 2z � 10 3x � 5y � 4z � 5

13. 2x � 3y � z � �9 14. 3x � 4y � z � 19 �2x � y � 3z � 7 2x � 4y � z � 0

x � y � 2z � �5 x � 2y � 5z � 17

15. 2x � 5y � 2z � 16 16. �2x � 3y � 4z � 3 3x � 2y � 3z � �19 3x � 5y � 2z � 4 4x � 3y � 4z � 18 �4x � 2y � 3z � 0

17. x � y � 4 18. x � y � z � 0 y � z � �2 x � y � �2

x � y � z � 9 y � z � 10

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 494

7-38 494 Chapter 7 Systems of Linear Equations

19. x + y = 7 20. 2x - y = -8 y - z = -1 y + 3z = 22

x + 3z = 18 x - z = -8

U3V Dependent and Inconsistent Systems Solve each system. See Examples 3 and 4.

21. x + y - z = 2 22. x + y + z = 1 -x - y + z = -2 2x + 2y + 2z = 2 2x + 2y - 2z = 4 4x + 4y + 4z = 4

23. x + y - z = 2 24. x + y + z = 6 x + y + z = 8 2x + 2y + 2z = 9 x + y - z = 6 3x + 3y + 3z = 12

25. x + y + z = 9 26. x - y + z = 2 x + y = 5 y - z = 3

z = 1 x = 4

27. x - y + 2z = 3 28. 4x - 2y - 2z = 5 2x + y - z = 5 2x - y - z = 7 3x - 3y + 6z = 4 -4x + 2y + 2z = 6

29. 2x - 4y + 6z = 12 30. 3x - y + z = 5 6x - 12y + 18z = 36 9x - 3y + 3z = 15

-x + 2y - 3z = -6 -12x + 4y - 4z = -20

31. x - y = 3 32. 2x - y = 6 y + z = 8 2y + z = -4

2x + 2z = 7 8x + 2z = 3

33. 0.10x + 0.08y - 0.04z = 3 5x + 4y - 2z = 150

0.3x + 0.24y - 0.12z = 9

34. 0.06x - 0.04y + z = 6 3x - 2y + 50z = 300

0.03x - 0.02y + 0.5z = 3

Use a calculator to solve each system.

35. 3x + 2y - 0.4z = 0.1 3.7x - 0.2y + 0.05z = 0.41 -2x + 3.8y - 2.1z = -3.26

36. 3x - 0.4y + 9z = 1.668 0.3x + 5y - 8z = -0.972

5x - 4y - 8z = 1.8

Use a calculator to check whether the given ordered triple satisfies all three equations of the given system.

37. (45, 32, 12) 3x - 2y + z = 83

x + 5y - z = 193 5x + y - 6z = 185

38. (-16, 45, 19) 7x + 6y - 3z = 101 3x + 4y - 9z = -39

-x + 5y + 8z = 393

39. (244, 386, 122) 0.1x + 0.3y - 0.12z = 125.56 0.9x + 0.4y - 0.25z = 343.5 0.5x + 0.2y + 0.15z = 181.0

40. (66, 72, 84) 1 1 5 ��x + ��y - ��z = -3 2 1

-��x - 3 5 ��x - 6

3 7 1 5 ��y + ��z = -5 4 12 5 5 ��y + �� z = 5 4 14

Miscellaneous Solve each system. State whether the system is independent, dependent, or inconsistent.

41.

42.

43.

44.

45.

46.

x - 2y = -12 2x + 3y = 4

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

x + y = 4 2x + 2y = 8

-x + y = 12 5x + 4y = -6

x + 2y - 3z = 6 -2x - 4y + 6z = 10

x - y - z = 4 2x + y + 3z = 6 2x - 2y - 2z = 10

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 495

7-39 7.4 Systems of Linear Equations in Three Variables 495

47. x - y – z = 0 x + y + 3z = 8 x + 3y + z =10

48. x + y - z = 0 3x - y - z = -10 2x + y + z = 35

49. 5x - 5y + 5z = 5 x - y + z = 1

3x - 3y + 3z = 3

50. 4x - 7y + 3z = 5 8x - 14y + 6z = 10

U4V Applications

Solve each problem by using a system of three equations in three unknowns. See Example 5.

51. Three cars. The town of Springfield purchased a Chevrolet, a Ford, and a Toyota for a total of $66,000. The Ford was $2000 more than the Chevrolet and the Toyota was $2000 more than the Ford. What was the price of each car?

52. Buying texts. Melissa purchased an English text, a math text, and a chemistry text for a total of $276. The English text was $20 more than the math text, and the chemistry text was twice the price of the math text. What was the price of each text?

53. Three-day drive. In three days, Carter drove 2196 miles in 36 hours behind the wheel. The first day he averaged 64 mph, the second day 62 mph, and the third day 58 mph. If he drove 4 more hours on the third day than on the first day, then how many hours did he drive each day?

54. Three-day trip. In three days, Katy traveled 146 miles down the Mississippi River in her kayak with 30 hours of paddling. The first day she averaged 6 mph, the second day 5 mph, and the third day 4 mph. If her distance on the third day was equal to her distance on the first day, then for how many hours did she paddle each day?

55. Diversification. Ann invested a total of $12,000 in stocks, bonds, and a mutual fund. She received a 10% return on her stock investment, an 8% return on her bond invest­ ment, and a 12% return on her mutual fund. Her total return was $1230. If the total investment in stocks and

bonds equaled her mutual fund investment, then how much did she invest in each?

56. Paranoia. Fearful of a bank failure, Norman split his life savings of $60,000 among three banks. He received 5%, 6%, and 7% on the three deposits. In the account earning 7% interest, he deposited twice as much as in the account earning 5% interest. If his total earnings were $3760, then how much did he deposit in each account?

57. Weighing in. Anna, Bob, and Chris will not disclose their weights but agree to be weighed in pairs. Anna and Bob together weigh 226 pounds. Bob and Chris together weigh 210 pounds. Anna and Chris together weigh 200 pounds. How much does each student weigh?

226 210 200

Anna & Bob Bob & Chris Anna & Chris

Figure for Exercise 57

58. Big tipper. On Monday Headley paid $1.70 for two cups of coffee and one doughnut, including the tip. On Tuesday he paid $1.65 for two doughnuts and a cup of coffee, including the tip. On Wednesday he paid $1.30 for one coffee and one doughnut, including the tip. If he always tips the same amount, then what is the amount of each item?

59. Three coins. Nelson paid $1.75 for his lunch with 13 coins, consisting of nickels, dimes, and quarters. If the number of dimes was twice the number of nickels, then how many of each type of coin did he use?

60. Pocket change. Harry has $2.25 in nickels, dimes, and quarters. If he had twice as many nickels, half as many dimes, and the same number of quarters, he would have $2.50. If he has 27 coins altogether, then how many of each does he have?

61. Working overtime. To make ends meet, Ms. Farnsby works three jobs. Her total income last year was $48,000. Her income from teaching was just $6000 more than her income from house painting. Royalties from her textbook sales were one-seventh of the total money she received from teaching

dug84356_ch07b.qxd 9/14/10 1:57 PM Page 496

496 Chapter 7 Systems of Linear Equations 7-40

and house painting. How much did she make from each source last year?

62. Lunch-box special. Salvador’s Fruit Mart sells variety packs. The small pack contains three bananas, two apples, and one orange for $1.80. The medium pack contains four bananas, three apples, and three oranges for $3.05. The family size contains six bananas, five apples, and four oranges for $4.65. What price should Salvador charge for his lunch-box special that consists of one banana, one apple, and one orange?

63. Three generations. Edwin, his father, and his grandfather have an average age of 53. One-half of his grandfather’s age, plus one-third of his father’s age, plus one-fourth of Edwin’s age is 65. If 4 years ago, Edwin’s grandfather was four times as old as Edwin, then how old are they all now?

64. Three-digit number. The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the old number. If the hundreds digit plus twice the tens digit is equal to the units digit, then what is the number?

Getting More Involved

65. Exploration

Draw diagrams showing the possible ways to position three planes in three-dimensional space.

66. Discussion

Make up a system of three linear equations in three variables for which the solution set is {(0, 0, 0)}. A system with this solution set is called a homogeneous system. Why do you think it is given that name?

C h

a p

t e

r

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 497

7-41 Chapter 7 Enriching Your Mathematical Word Power 497

7 Wrap-Up Summary

Systems of Linear Equations Examples

Methods for solving Graphing: Sketch the graphs to see the solution. The graphs of systems in two y = x - 1 and variables x + y = 3 intersect

at (2, 1).

Substitution: Solve one equation for one variable Substitution: in terms of the other, and then substitute into the x + (x - 1) = 3 other equation.

Addition: Multiply each equation as necessary to -x + y = -1 eliminate a variable upon addition of the equations. x + y = 3

2y = 2

Types of linear systems Independent: One point in solution set y = x - 5 in two variables The lines intersect at one point. y = 2x + 3

Dependent: Infinite solution set 2x + 3y = 4 The lines are the same. 4x + 6y = 8

Inconsistent: Empty solution set 2x + y = 1 The lines are parallel. 2x + y = 5

Linear equation Ax + By + Cz = D 2x - y + 3z = 5 in three variables In a three-dimensional coordinate system the graph

is a plane.

Linear systems Use substitution or addition to eliminate variables in x + y - z = 3 in three variables the system. The solution set may be a single point, 2x - 3y + z = 2

the empty set, or an infinite set of points. x - y - 4z = 14

Enriching Your Mathematical Word Power

Fill in the blank. 5. In the method a variable is eliminated by 1. A(n) of equations consists of two or more equations. substituting one equation into the other.

2. A(n) linear system is a system with exactly 6. In the method a variable is eliminated by adding one solution. the equations.

3. A(n) system has no solutions. 7. A(n) equation in three variables has the form Ax + By + Cz = D with A, B, and C not all zero.4. A(n) system has infinitely many solutions.

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 498

7-42 498 Chapter 7 Systems of Linear Equations

Review Exercises

7.1 The Graphing Method Solve by graphing. Indicate whether each system is independent, dependent, or inconsistent.

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

2. y = 3x - 4 y = -2x + 1

3. x + 2y = 4 1

y = -�� x + 2 2

4. 2x - 3y = 12 3y - 2x = -12

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

6. 3x - y = 4 3x - y = 0

7.2 The Substitution Method

Solve by substitution. Indicate whether each system is independent, dependent, or inconsistent.

7. y = 3x + 11 2x + 3y = 0

8. x - y = 3 3x - 2y = 3

9. x = y + 5 2x - 2y = 12

10. 3y = x + 5 3x - 9y = -10

11. 2x - y = 3 6x - 9 = 3y

1 12. y = �� x - 9

2 3x - 6y = 54

1 13. y = �� x - 3

14.

15.

16.

2 1

y = �� x + 2 3

1 x = �� y - 1

8 1

y = �� x + 39 4

x + 2y = 1 8x + 6y = 4

x - 5y = 4 4x + 8y = -5

7.3 The Addition Method Solve by addition. Indicate whether each system is independent, dependent, or inconsistent.

17. 5x - 3y = -20 3x + 2y = 7

18. -3x + y = 3 2x - 3y = 5

19. 2(y - 5) + 4 = 3(x - 6) 3x - 2y = 12

20. x + 3(y - 1) = 11 2(x - y) + 8y = 28

21. 3x - 4(y - 5) = x + 2 2y - x = 7

22. 4(1 - x) + y = 3 3(1 - y) - 4x = -4y

1 3 3 23. �� x + �� y = ��

4 8 8 5 �� x - 6y = 7 2

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 499

7-43

1 1 1 24. �� x - �� y = ��

3 6 3 1 1 �� x + �� y = 0 6 4

25. 0.4x + 0.06y = 11.6 0.8x - 0.05y = 13

26. 0.08x + 0.7y = 37.4 0.06x - 0.05y = -0.7

7.4 Systems of Linear Equations in Three Variables Solve each system by elimination of variables.

27. x - y + z = 4 -x + 2y - z = 0 -x + y - 3z = -16

28. 2x - y + z = 5 x + y - 2z = -4

3x - y + 3z = 10

29. 2x - y - z = 3 3x + y + 2z = 4 4x + 2y - z = -4

30. 2x + 3y - 2z = -11 3x - 2y + 3z = 7

x - 4y + 4z = 14

31. x + y - z = 4 y + z = 6

x + 2y = 8

32. x - 3y + z = 5 2x - 4y - z = 7 2x - 6y + 2z = 6

33. x - 2y + z = 8 -x + 2y - z = -8 2x - 4y + 2z = 16

34. x - y + z = 1 2x - 2y + 2z = 2

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

Chapter 7 Review Exercises 499

Miscellaneous Solve each system by the method of your choice.

35. x + y = 7 -x + 2y = 5

37. 2x + y = 0 x - 3y = 14

39. 2x - y = 0 3x + y = -5

41. y = 2x - 3 3x - 2y = 4

43. x + y - z = 0 x - y + 2z = 4

2x + y - z = 1

45. x + y = 3 x + y + z = 0 x - y - z = 2

47. y = 2x - 30 1 1 �� x - �� y = -1 5 2

49. 2x + y = 9 2x - 5y = 15

51. x - y = 0 2x + 3y = 35

53. x + y = 40 0.2x + 0.8y = 23

55. y = 2x - 5 y + 1 = 2(x - 2)

57. x - y = 5 2x = 2y + 14

5 59. y = �� x

7 2

x = -�� y 3

36. -x + y = 1 2x - 3y = -7

38. 2x - y = 8 3x + 2y = -2

40. 3x - 2y = 14 2x + 3y = -8

42. y = 2x - 5 y = 3x - 3y

44. 2x - y + 2z = 9 x + 3y = 5

3x + z = 9

46. 2x - y + z = 0 4x + 6y - 2z = 0

x - 2y - z = -9

48. 3x - 5y = 4 3

y = �� x - 2 4

50. 3y - x = 0 x - 4y = -2

52. 2y = x + 6 -3x + 2y = -2

54. x - y = 10 0.1x + 0.5y = 13

56. 2x - y = 3 2y = 4x - 6

58. 2x - y = 4 2x - y = 3

60. 7y = 9x

-3x = 4y

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 500

500 Chapter 7 Systems of Linear Equations 7-44

61. 3(y - 1) = 2(x - 3) 62. y = 3(x - 4) 3y - 2x = -3 3x - y = 12

63. y = 3x 64. y = 3x - 4 y = 3x + 1 y = 3x + 4

65. x - y = 0.1 66. y - 2x = -7.5 2x - 3y = -0.5 3x - 5y = 3.2

67. y = 2x + 4 68. 3x - 2y = 6 3x + y = -1 3x + 2y = 6

1 69. y = -��x + 4 70. 2x - 3y = 6

2 2

x + 2y = 8 y = ��x - 2 3

71. 2y - 2x = 2 72. 3y - 3x = 9 2y - 2x = 6 x - y = 1

1 2 73. y = -�� x 74. y = -�� x

4 3 x + 4y = 8 2x + 3y = 5

Use a system of equations in two or three variables to solve each problem. Solve by the method of your choice.

75. Perimeter of a rectangle. The length of a rectangular swimming pool is 15 feet longer than the width. If the perimeter is 82 feet, then what are the length and width?

76. Household income. Alkena and Hsu together earn $84,326 per year. If Alkena earns $12,468 more per year than Hsu, then how much does each of them earn per year?

77. Two-digit number. The sum of the digits in a two-digit number is 15. When the digits are reversed, the new number is 9 more than the original number. What is the original number?

78. Two-digit number. The sum of the digits in a two-digit number is 8. When the digits are reversed, the new number is 18 less than the original number. What is the original number?

79. Traveling by boat. Alonzo can travel from his camp downstream to the mouth of the river in 30 minutes. If it takes him 45 minutes to come back, then how long would it take him to go that same distance in the lake with no current?

Time with current � 30 min Time against current � 45 min

Figure for Exercise 79

80. Driving and dating. In 4 years Gasper will be old enough to drive. His parents said that he must have a driver’s license for 2 years before he can date. Three years ago, Gasper’s age was only one-half of the age necessary to date. How old must Gasper be to drive, and how old is he now?

81. Three solutions. A chemist has three solutions of acid that must be mixed to obtain 20 liters of a solution that is 38% acid. Solution A is 30% acid, solution B is 20% acid, and solution C is 60% acid. Because of another chemical in these solutions, the chemist must keep the ratio of solution C to solution A at 2 to 1. How many liters of each should she mix together?

82. Mixing investments. Darlene invested a total of $20,000. The part that she invested in Dell Computer stock returned 70%, and the part that she invested in U.S. Treasury bonds returned 5%. Her total return on these two investments was $9580.

a) Use the accompanying graph to estimate the amount that she put into each investment.

b) Solve a system of equations to find the exact amount that she put into each investment.

A m

ou nt

in b

on ds

(i n

th ou

sa nd

s of

d ol

la rs

)

150

0

50

100

0 10 15 5 20 Amount in Dell

(in thousands of dollars)

Total return

Total investment

Figure for Exercise 82

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 501

7-45

83. Beets and beans. One serving of canned beets contains 1 gram of protein and 6 grams of carbohydrates. One serving of canned red beans contains 6 grams of protein and 20 grams of carbohydrates. How many servings of each would it take to get exactly 21 grams of protein and 78 grams of carbohydrates?

84. Protein and carbohydrates. One serving of Cornies breakfast cereal contains 2 grams of protein and 25 grams of carbohydrates. One serving of Oaties breakfast cereal contains 4 grams of protein and 20 grams of carbohydrates. How many servings of each would provide exactly 24 grams of protein and 210 grams of carbohydrates?

85. Milk and a magazine. Althia bought a gallon of milk and a magazine for a total of $4.65, excluding tax. Including the tax, the bill was $4.95. If there is a 5% sales tax on milk and an 8% sales tax on magazines, then what was the price of each item?

86. Rectangular patio. The length of a rectangular patio is 12 feet greater than the width. If the perimeter is 84 feet, then what are the length and width?

87. Rectangular notepad. The length of a rectangular notepad is 2 cm longer than twice the width. If the perimeter is 34 cm, then what are the length and width?

88. Rectangular table. The width of a rectangular table is 1 ft less than half of the length. If the perimeter is 28 ft, then what are the length and width?

89. Rectangular painting. The width of a rectangular painting is two-thirds of its length. If the perimeter is 60 in., then what are the length and width?

90. Sum and difference. The sum of two numbers is 10 and their difference is 3. Find the numbers.

91. Sum and difference. The sum of two numbers is 51 and their difference is 26. Find the numbers.

92. Sum and difference. The sum of two numbers is 1 and their difference is 20. Find the numbers.

93. Sum and difference. The sum of two numbers is 5 and their difference is 30. Find the numbers.

94. Washing machines and refrigerators. A truck carrying 3600 cubic feet of cargo consisting of washing machines and refrigerators was hijacked. The washing machines are

Chapter 7 Review Exercises 501

worth $300 each and are shipped in 36-cubic-foot cartons. The refrigerators are worth $900 each and are shipped in 45-cubic-foot cartons. If the total value of the cargo was $51,000, then how many of each were there on the truck?

95. Parking lot boredom. A late-night parking lot attendant counted 50 vehicles on the lot consisting of four-wheel cars, three-wheel cars, and two-wheel motorcycles. She then counted 192 tires touching the ground and observed that the number of four-wheel cars was nine times the total of the other vehicles on the lot. How many of each type of vehicle were on the lot?

96. Happy meals. The total price of a hamburger, an order of fries, and Coke at a fast-food restaurant is $3.00. The price of a hamburger minus the price of an order of fries is $0.20 and the price of an order of fries minus the price of a Coke is also $0.20. Find the price of each item.

97. Singles and doubles. Windy’s Hamburger Palace sells singles and doubles. Toward the end of the evening, Windy himself noticed that he had on hand only 32 patties and 34 slices of tomatoes. If a single takes l patty and 2 slices, and a double takes 2 patties and 1 slice, then how many more singles and doubles must Windy sell to use up all of his patties and tomato slices?

98. Valuable wrenches. Carmen has a total of 28 wrenches, all of which are either box wrenches or open-end wrenches. For insurance purposes she values the box wrenches at $3.00 each and the open-end wrenches at $2.50 each. If the value of her wrench collection is $78, then how many of each type does she have?

99. Gary and Harry. Gary is 5 years older than Harry. Twenty-nine years ago, Gary was twice as old as Harry. How old are they now?

100. Acute angles. One acute angle of a right triangle is 3° more than twice the other acute angle. What are the sizes of the acute angles?

x

2x � 3

Figure for Exercise 100

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 502

502 Chapter 7 Systems of Linear Equations 7-46

101. Equal perimeters. A rope of length 80 feet is to be cut 105. Weighing dogs. Cassandra wants to determine the into two pieces. One piece will be used to form a square, weights of her two dogs, Mimi and Mitzi. However, and the other will be used to form an equilateral triangle. neither dog will sit on the scale by herself. Cassandra, If the figures are to have equal perimeters, then what Mimi, and Mitzi altogether weigh 175 pounds. Cassandra should be the length of a side of each? and Mimi together weigh 143 pounds. Cassandra and

Mitzi together weigh 139 pounds. How much does each weigh individually?

Figure for Exercise 101

102. Coffee and doughnuts. For a cup of coffee and a dough- nut, Thurrel spent $2.25, including a tip. Later he spent $4.00 for two coffees and three doughnuts, including a tip. If he always tips $1.00, then what is the price of a cup of coffee?

103. Chlorine mixture. A 10% chlorine solution is to be mixed with a 25% chlorine solution to obtain 30 gallons of 20% solution. How many gallons of each must be used?

104. Safe drivers. Emily and Camille started from the same city and drove in opposite directions on the freeway. After 3 hours, they were 354 miles apart. If they had gone in the same direction, Emily would have been 18 miles ahead of Camille. How fast did each woman drive?

175 143 139

Cassandra Cassandra Cassandra Mimi Mimi Mitzi Mitzi

Figure for Exercise 105

106. Nickels, dimes, and quarters. Bernard has 41 coins consisting of nickels, dimes, and quarters, and they are worth a total of $4.00. If the number of dimes plus the number of quarters is one more than the number of nickels, then how many of each does he have?

107. Finding three angles. If the two acute angles of a right triangle differ by 12°, then what are the measures of the three angles of this triangle?

108. Two acute and one obtuse. The obtuse angle of a trian­ gle is twice as large as the sum of the two acute angles. If the smallest angle is only one-eighth as large as the sum of the other two, then what is the measure of each angle?

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 503

7-47 Chapter 7 Test 503

Chapter 7 Test

Solve the system by graphing.

1. x + y = 4 y = 2x + 1

Solve each system by substitution.

2. y = 2x - 8 3. y = x - 5 4x + 3y = 1 3x - 4(y - 2) = 28 - x

Solve each system by the addition method.

4. 3x + 2y = 3 5. 3x - y = 5 4x - 3y = -13 -6x + 2y = 1

Determine whether each system is independent, dependent, or inconsistent.

6. y = 3x - 5 7. 2x + 2y = 8 y = 3x + 2 x + y = 4

8. y = 2x - 3 y = 5x - 14

Solve each system by the method of your choice.

9. 3x - y = 1 10. 2x - y = -4 x + 2y = 12 3x + y = -1

11. x + y = 0 x - y + 2z = 6

2x + y - z = 1

12. x + y - z = 2 2x - y + 3z = -5 x - 3y + z = 4

13. x - y - z = 1 -x - y + 2z = -2 -x - 3y + z = -5

For each problem, write a system of equations in two or three variables. Use the method of your choice to solve each system.

14. One night the manager of the Sea Breeze Motel rented 5 singles and 12 doubles for a total of $1583. The next night he rented 9 singles and 10 doubles for a total of $1701. What is the rental charge for each type of room?

15. Jill, Karen, and Betsy studied a total of 93 hours last week. Jill’s and Karen’s study time totaled only one-half as much as Betsy’s. If Jill studied 3 hours more than Karen, then how many hours did each one of the girls spend studying?

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 504

504 Chapter 7 Systems of Linear Equations 7-48

MakingConnections A Review of Chapters 1–7

Simplify each expression.

1. -34

1 2. �� (3) + 6

3

3. (-5)2 - 4(-2)(6)

4. 6 - (0.2)(0.3)

5. 5(t - 3) - 6(t - 2)

6. 0.1(x - 1) - (x - 1) 2-9x - 6x + 3

7. �� -3

4y - 6 3y - 9 8. � - �

2 3

Factor each polynomial completely.

9. 3y3 - 363y

10. 2y4 - 32

11. yw + 2w - 4y - 8

12. y3 - 27

13. -3y2 - 12y + 135 314. -24y - 2y2 + 12y

15. 4a3 - 4a2 + 12a

16. 2a3b3 + 2ab5

Reduce each rational expression to lowest terms. 318x

17. �442x 812x

18. � 18x

2x + 8 19. �

2x � 14 2 2x - y

20. � x2 � xy

2 x - x - 30 21. ��2x - 5x - 6

22x + 9x + 4 22. ��

2x2 � x � 1

Perform the indicated operations.

1 3 23. �� + ��

6 8

4 3 24. �� + ��

15 20 1 1

25. �� - �� 5 12 3 1

26. �� - �� 10 6 2 21

27. �� � �� 15 22 3

28. �� � 88 4 2

29. �� � 4 5 9 3

30. �� � �� 20 10 1 5

31. �� + ��23a 6a

1 32. �� + y

2y

3 33. �� � (3x - 9)2x - 9

5ab6 14x3y7 34. �� � ��

7x3y5 15ab

6a 35. �� � a

5b 2a - 4 2a - 4

36. �� � �� a2 + 8a + 12 a2 - 36

Solve each equation for y.

37. 3x - 5y = 7

38. Cx - Dy = W

39. Cy = Wy - K

1 40. A = �� b(w - y)

2

Solve each system.

41. y = x - 5 2x + 3y = 5

42. 0.05x + 0.06y = 67 x + y = 1200

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 505

7-49 Chapter 7 Making Connections 505

43. 3x - 15y = -51 x + 17 = 5y

44. 0.07a + 0.3b = 6.70 7a + 30b = 67

Find the equation of each line.

45. The line through (0, 55) and (-99, 0)

46. The line through (2, -3) and (-4, 8)

47. The line through (-4, 6) that is parallel to y = 5x

Solve.

51. Comparing copiers. A self-employed consultant has prepared the accompanying graph to compare the total cost of purchasing and using two different copy machines. a) Which machine has the larger purchase price? b) What is the per copy cost for operating each machine,

not including the purchase price? c) Find the slope of each line and interpret your findings. d) Find the equation of each line. e) Find the number of copies for which the total cost is the

same for both machines.

48. The line through (4, 7) that is perpendicular to y = -2x + 1

49. The line through (3, 5) that is parallel to the x-axis

50. The line through (-7, 0) that is perpendicular to the x-axis C

os t

(i n

th ou

sa nd

s of

d ol

la rs

) $14,000 14 12

Machine A

Machine B

10 8 6 4

$13,000

2

0 100 200 300 Number of copies

(in thousands)

Figure for Exercise 51

dug84356_ch07c.qxd 9/14/10 1:59 PM Page 506

506 Chapter 7 Systems of Linear Equations 7-50

CriticalThinking For Individual or Group Work Chapter 7

These exercises can be solved by a variety of techniques, which may or may not require algebra. So be creative and think critically. Explain all answers. Answers are in the Instructor’s Edition of this text.

1. Tricky square. Start with a square and write any integer at each vertex. (a) At the midpoint of each side write the absolute value of the difference between the numbers at the endpoints of that side. (b) Connect the midpoints to obtain another square. Repeat parts (a) and (b) to obtain a sequence of nested squares as shown in the accompanying figure. What numbers will you always end up with?

Figure for Exercise 1

2. Planning ahead. Thaddeus takes one month to build a kayak (K) and two months to build a canoe (C).

Photo for Exercise 2

While planning ahead for 1 month, he notes that there is only one thing to do that will not result in any partially finished boats. That is, build one kayak. For planning 2 months ahead there are two possibilities, KK or C. For a 3-month plan there are three possibilities, KKK or KC or CK.

a) Find the number of possibilities for a 4-month plan, a 5-month plan, and a 6-month plan by listing the possibilities. Look for a pattern.

b) Find the number of possibilities for a 7-month plan and an 8-month plan without making a list.

3. Five coins. Place five coins on a table with heads facing downward. On each move you must turn over exactly three coins. What is the minimum number of moves necessary to get all five heads facing upward?

4. Rotating tires. Helen bought a new car with four tires and a full-size spare. If she rotated the tires so that each tire would have the same amount of wear, then how many miles were on each tire when her odometer showed 40,000 miles?

5. Cutting pizza. What is the largest number of pieces of pizza you can get if you cut a circular pizza with five straight cuts? What is the largest number of pieces of pizza you can get if you cut a circular pizza with seven straight cuts?

6. Mysterious rectangle. The length of a rectangle is a two- digit number with identical digits (aa). The width of the rectangle is one-tenth of the length (a.a). The perimeter is numerically twice as large as the area. Find the length, width, perimeter, and area.

7. Finding squares. Evaluate the expression

1002 - 992 + 982 - 972 + 962 - � � � - 32 + 22 - 12

without using a calculator.

8. Five-digit sum. Find the sum of all five-digit numbers that are formed by using the digits 1, 2, 3, 4, and 5 once and only once.