7.2 the substitution method answers

C h

a p

t e

r

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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)

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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.

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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.

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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

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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

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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

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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

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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

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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.

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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

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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.

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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

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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.

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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

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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

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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

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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?

http:www.movieweb.com
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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.

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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.

http:www.cdc.gov