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E. Inconsistent and Dependent Systems

E. Inconsistent and Dependent Systems

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The solutions of a dependent system are often written in set notation as the set

of ordered pairs (x, y), where y is a specified function of x. Here the solution would

be 5 1x, y2 0 y ϭ Ϫ34x ϩ 36. Using an ordered pair with an arbitrary variable, called a

Ϫ3p

ϩ 3b.

parameter, is also common: ap,

4

Now try Exercises 49 through 60 ᮣ



Figure 6.7

5



Ϫ5



5



Ϫ5



Figure 6.8



If we had attempted to solve the system in Example 6

Figure 6.6

by graphing (Figure 6.6), we could be mislead into

thinking something is wrong—because only one line

is visible (Figure 6.7). In this case, using the TABLE

feature of the calculator would help verify that the system is dependent. Since the ordered pair solutions are

identical (try scrolling through positive and negative

values), the equations must be dependent (Figure 6.8).

Finally, if the lines have equal slopes and different

y-intercepts, they are parallel and the system will have no solution. A system with no

solutions is called an inconsistent system. An “inconsistent system” produces an “inconsistent answer,” such as 12 ϭ 0 or some other false statement when substitution or

elimination is applied. In other words, all variable terms are once again eliminated, but

the remaining statement is false. A summary of the three possibilities is shown in Figure 6.9 for arbitrary slope m and y-intercept (0, b).



Figure 6.9

Independent

m1 ϭ m2



Dependent

m1 ϭ m2, b1 ϭ b2



y



E. You’ve just seen how

we can recognize inconsistent

systems and dependent

systems



Inconsistent

m1 ϭ m2, b1 ϭ b2



y



x



One point in common



y



x



All points in common



x



No points in common



F. Systems and Modeling

In previous chapters, we solved numerous real-world applications by writing all given

relationships in terms of a single variable. Many situations are easier to model using a

system of equations with each relationship modeled independently using two variables. We begin here with a mixture application. Although they appear in many different forms (coin problems, metal alloys, investments, merchandising, and so on),

mixture problems all have a similar theme. Generally one equation is related to quantity (how much of each item is being combined) and one equation is related to value

(what is the value of each item being combined).

EXAMPLE 7







Solving a Mixture Application

A jeweler is commissioned to create a piece of artwork that will weigh 14 oz and

consist of 75% gold. She has on hand two alloys that are 60% and 80% gold,

respectively. How much of each should she use?



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Solution







WORTHY OF NOTE

As an estimation tool, note that if

equal amounts of the 60% and

80% alloys were used (7 oz each),

the result would be a 70% alloy

(halfway in between). Since a

75% alloy is needed, more of the

80% gold will be used.



Y1 ϭ 14 Ϫ X, Y2 ϭ



Let x represent ounces of the 60% alloy and y represent ounces of the 80% alloy.

The first equation must be x ϩ y ϭ 14, since the piece of art must weigh exactly

14 oz (this is the quantity equation). The x ounces are 60% gold, the y ounces are

80% gold, and the 14 oz will be 75% gold. This gives the value equation:

x ϩ y ϭ 14

0.6x ϩ 0.8y ϭ 0.751142. The system is e

(after clearing decimals).

6x ϩ 8y ϭ 105

Solving for y in the first equation gives y ϭ 14 Ϫ x. Substituting 14 Ϫ x for y in

the second equation gives



105 Ϫ 6X

8



6x ϩ 8y ϭ 105

6x ϩ 8114 Ϫ x2 ϭ 105

6x ϩ 112 Ϫ 8x ϭ 105

Ϫ2x ϩ 112 ϭ 105

7



2



15



Ϫ5



583



15



second equation

substitute 14 Ϫ x for y

distribute

simplify

solve for x



Substituting 72 for x in the first equation gives y ϭ 21

2 . She should use 3.5 oz of the

60% alloy and 10.5 oz of the 80% alloy. A graphical check is shown in the figure.



Ϫ5



Now try Exercises 63 through 70 ᮣ

A second example involves an application of uniform motion (distance ϭ

rate # time), and explores concepts of great importance to the navigation of ships and

airplanes. As a simple illustration, if you’ve ever walked at your normal rate r on the

“moving walkways” at an airport, you likely noticed an increase in your total speed.

This is because the resulting speed combines your walking rate r with the speed w of

the walkway: total speed ϭ r ϩ w. If you walk in the opposite direction of the walkway, your total speed is much slower, as now total speed ϭ r Ϫ w.

This same phenomenon is observed when an airplane is flying with or against the

wind, or a ship is sailing with or against the current.

EXAMPLE 8







Solving an Application of Systems—Uniform Motion

An airplane flying due south from St. Louis, Missouri, to Baton Rouge, Louisiana,

uses a strong, steady tailwind to complete the trip in only 2.5 hr. On the return trip,

the same wind slows the flight and it takes 3 hr to get back. If the flight distance

between these cities is 912 km, what is the cruising speed of the airplane (speed

with no wind)? How fast is the wind blowing?



Solution







Let r represent the rate of the plane and w the rate of the wind. Since D ϭ RT, the

flight to Baton Rouge can be modeled by 912 ϭ 1r ϩ w2 12.52 , and the return flight

by 912 ϭ 1r Ϫ w2132 . This produces the system e







Algebraic Solution



Dividing R1 by 2.5 and R2 by 3 produces

the following sequence:

R1

364.8 ϭ r ϩ w

2.5 912 ϭ 2.5r ϩ 2.5w

e

Se

R2 912 ϭ 3r Ϫ 3w

304.0 ϭ r Ϫ w

3







912 ϭ 2.5r ϩ 2.5w

.

912 ϭ 3r Ϫ 3w



Graphical Solution



Using x for w and y for r, we solve each equation for y and

obtain:

Y1 ϭ



912 Ϫ 2.5X

2.5



Y2 ϭ



912 ϩ 3X

3



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Using R1 ϩ R2 gives 668.8 ϭ 2r, showing

334.4 ϭ r. The speed of the plane is 334.4 kph.

Substituting 334.4 for r in the second equation,

we have:

912 ϭ 3r Ϫ 3w

912 ϭ 31334.42 Ϫ 3w

912 ϭ 1003.2 Ϫ 3w

Ϫ91.2 ϭ Ϫ3w

30.4 ϭ w



We then set an appropriate window and graph these equations

to find the point of intersection.

400



equation

substitute

multiply



0



50



subtract 1003.2

divide by Ϫ3



The speed of the wind is 30.4 kph.



200



The speed of the wind (x) is 30.4 kph, and the speed of the

plane (y) is 334.4 kph.

Now try Exercises 71 through 74 ᮣ



Systems of equations also play a significant role in cost-based pricing in the business world. The costs involved in running a business can broadly be understood as

either a fixed cost k or a variable cost v. Fixed costs might include the monthly rent

paid for facilities, which remains the same regardless of how many items are produced

and sold. Variable costs would include the cost of materials needed to produce the item,

which depends on the number of items made. The total cost can then be modeled by

C1x2 ϭ vx ϩ k for x number of items. Once a selling price p has been determined, the

revenue equation is simply R1x2 ϭ px (price times number of items sold). We can now

set up and solve a system of equations that will determine how many items must be sold

to break even, performing what is called a break-even analysis where C(x) ϭ R(x).

EXAMPLE 9







Solving an Application of Systems: Break-Even Analysis

In home businesses that produce items to sell

on Ebay®, fixed costs are easily determined by

rent and utilities, and variable costs by the price

of materials needed to produce the item.

Karen’s home business makes large decorative

candles for all occasions. The cost of materials

is $3.50 per candle, and her rent and utilities

average $900 per month. If her candles sell for

$9.50, how many candles must be sold each

month to break even?



Solution







Let x represent the number of candles sold. Her total cost is C1x2 ϭ 3.5x ϩ 900

(variable cost plus fixed cost), and projected revenue is R1x2 ϭ 9.5x. This gives the

C1x2 ϭ 3.5x ϩ 900

system e

. To break even, Cost ϭ Revenue which gives

R 1x2 ϭ 9.5x

9.5x ϭ 3.5x ϩ 900

6x ϭ 900

x ϭ 150



The analysis shows that Karen must sell 150 candles each month to break even.

Now try Exercises 75 through 78 ᮣ



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WORTHY OF NOTE

There are limitations to this model,

and this interplay can be affected

by the number of available

consumers, production limits,

“shelf-life” issues, and so on, but

at any given moment in the life

cycle of a product, consumer

demand responds to price in this

way. Producers also respond to

price in a very predictable way.



EXAMPLE 10







In a “free-market” economy, also referred to as a “supply-and-demand” economy, there are naturally occurring forces that invariably come into play if no outside forces act on the producers (suppliers) and consumers (demanders). Generally

speaking, the higher the price of an item, the lower the demand. A good advertising

campaign can increase the demand, but the increasing demand brings an increase in

price, which moderates the demand — and so it goes until a balance is reached.

These free-market forces ebb and flow until market equilibrium occurs, at the specific price where the supply and demand are equal.

In Exercises 75 to 78, the equation models were artificially constructed to yield a

“nice” solution. In actual practice, the equations and coefficients are not so “well

behaved” and are based on the collection and interpretation of real data. While market

analysts have sophisticated programs and numerous models to help develop these

equations, here we’ll use our experience with regression to develop the supply and

demand curves.

Using Technology to Find Market Equilibrium

A manufacturer of MP3 players has hired a

consulting firm to do market research on their

“next-generation” player. Over a 10-week period,

the firm collected the data shown for the MP3

player market (data includes MP3 players sold

and expected to sell).

a. Use a graphing calculator to simultaneously

display the demand and supply scatterplots.

b. Calculate a line of best fit for each and graph

them with the scatterplots (identify each curve).

c. Find the equilibrium point.



Solution



Figure 6.10



Figure 6.12







Price

Supply

(dollars) Demand (Inventory)

107.10



6900



12,200



85.50



7900



9900



64.80



13,200



8000



52.20



13,500



7900



108.00



6700



14,000



91.80



7600



12,000



77.40



9200



9400



46.80



13,800



6100



a. Begin by clearing all lists. This can be done

74.70 10,600

8800

manually, or by pressing 2nd + (MEM)

68.40 12,800

8600

and selecting option 4:ClrAllLists (the

command appears on the home screen).

Figure 6.11

Pressing

will execute the command,

16,000

and the word DONE will appear.

Carefully input price in L1, demand in

L2, and supply in L3 (see Figure 6.10).

With the window settings given in

40

115

Figure 6.11, pressing GRAPH will display

the price/demand and price/supply

scatterplots shown. If this is not the case,

use 2nd Y= (STAT PLOT) to be sure

3000

that “On” is highlighted in Plot1 and

Plot2, and that Plot1 uses L1 and L2, while Plot2 uses L1 and L3 (Figure 6.12).

Note we’ve chosen a different mark to indicate the data points in Plot2.

b. Calculate the linear regression equation for L1 and L2 (demand), and paste it in

Y1: LinReg (ax ؉ b) L1, L2, Y1 . Next, calculate the linear regression for

(recall that

L1 and L3 (supply) and paste it in Y2: LinReg (ax ؉ b) L1, L3, Y2

Y1 and Y2 are accessed using the VARS key). The resulting equations and graphs

are shown in Figures 6.13 and 6.14.

c. Once again we use 2nd TRACE (CALC) 5:intersect to find the equilibrium

point, which is approximately (80, 9931). Supply and demand for this MP3

player model are approximately equal at a price of about $80, with 9931 MP3

players bought and sold.

ENTER



ENTER



ENTER



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Figure 6.14

Figure 6.13



16,000



40



115



3000



Now try Exercises 79 through 82 ᮣ

F. You’ve just seen how

we can use a system of

equations to model and solve

applications



Other interesting applications can be found in the Exercise set. See Exercises 83

through 88.



6.1 EXERCISES





CONCEPTS AND VOCABULARY



Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.







1. Systems that have no solution are called

systems.



2. Systems having at least one solution are called

systems.



3. If the lines in a system intersect at a single point,

the system is said to be

and

.



4. If the lines in a system are coincident, the system is

referred to as

and

.



5. The given systems are equivalent. How do we

obtain the second system from the first?

2

1

5

xϩ yϭ

4x ϩ 3y ϭ 10

2

3

• 3

e

2x ϩ 4y ϭ 10

0.2x ϩ 0.4y ϭ 1



6. For e



2x ϩ 5y ϭ 8

,

3x ϩ 4y ϭ 5

which solution method would be more efficient,

substitution or elimination? Discuss/Explain why.



DEVELOPING YOUR SKILLS



Show the lines in each system would intersect in a single

point by writing the equations in slope-intercept form.



7x Ϫ 4y ϭ 24

7. e

4x ϩ 3y ϭ 15



0.3x Ϫ 0.4y ϭ 2

8. e

0.5x ϩ 0.2y ϭ Ϫ4



An ordered pair is a solution to an

equation if it makes the equation

true. Given the graph shown here,

determine which equation(s) have

the indicated point as a solution. If

the point satisfies more than one

equation, write the system for

which it is a solution.



9. A



10. B



y

5



3x ϩ 2y ϭ 6



yϭxϩ2

A



F



B



E

Ϫ5



5 x



C

x ϩ 3y ϭ Ϫ3

Ϫ5



D



11. C



12. D



13. E



14. F



Substitute the x- and y-values indicated by the ordered

pair to determine if it is a solution to the system. Also

check using the ALPHA keys on the home screen of a

graphing calculator.



15. e



3x ϩ y ϭ 11

13, 22

Ϫ5x ϩ y ϭ Ϫ13;



16. e



3x ϩ 7y ϭ Ϫ4

1Ϫ6, 22

7x ϩ 8y ϭ Ϫ21;



17. e



8x Ϫ 24y ϭ Ϫ17

7 5

aϪ , b

12x ϩ 30y ϭ 2;

8 12



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18. e



4x ϩ 15y ϭ 7

1 1

a , b

8x ϩ 21y ϭ 11; 2 3



Solve using elimination. In some cases, the system must

first be written in standard form. Verify solutions using

a graphing calculator.



Solve each system by graphing manually. Check results

by graphing the system on a graphing calculator, and

locating any points of intersection.



3x ϩ 2y ϭ 12

19. e

xϪ yϭ9



5x ϩ 2y ϭ Ϫ2

20. e

Ϫ3x ϩ y ϭ 10



5x Ϫ 2y ϭ 5

21. e

x ϩ 3y ϭ Ϫ16



3x ϩ y ϭ Ϫ2

22. e

5x ϩ 3y ϭ 2



Solve each system using substitution. Write solutions as

an ordered pair, and verify solutions using a graphing

calculator.



23. e



x ϭ 5y Ϫ 9

x Ϫ 2y ϭ Ϫ6



24. e



4x Ϫ 5y ϭ 7

2x Ϫ 5 ϭ y



25. e



y ϭ 23x Ϫ 7

3x Ϫ 2y ϭ 19



26. e



2x Ϫ y ϭ 6

y ϭ 34x Ϫ 1



Identify the equation and variable that makes the

substitution method easiest to use, then solve the system.

Verify solutions using a graphing calculator.



3x ϩ 2y ϭ 19

x Ϫ 4y ϭ Ϫ3



27. e



3x Ϫ 4y ϭ 24

5x ϩ y ϭ 17



29. e



0.7x ϩ 2y ϭ 5

0.8x ϩ y ϭ 7.4

30. e

0.6x ϩ 1.5y ϭ 9.3

x Ϫ 1.4y ϭ 11.4



31. e



5x Ϫ 6y ϭ 2

x ϩ 2y ϭ 6



28. e



32. e



2x ϩ 5y ϭ 5

8x Ϫ y ϭ 6



The substitution method can be used for like variables

or for like expressions. Solve the following systems, using

the expression common to both equations (do not solve

for x or y alone).



33. e



2x ϩ 4y ϭ 6

x ϩ 12 ϭ 4y



34. e



8x ϭ 3y ϩ 24

8x Ϫ 5y ϭ 36



35. e



5x Ϫ 11y ϭ 21

11y ϭ 5 Ϫ 8x



36. e



Ϫ6x ϭ 5y Ϫ 16

5y Ϫ 6x ϭ 4







587



37. e



2x Ϫ 4y ϭ 10

3x ϩ 4y ϭ 5



38. e



Ϫx ϩ 5y ϭ 8

x ϩ 2y ϭ 6



39. e



4x Ϫ 3y ϭ 1

3y ϭ Ϫ5x Ϫ 19



40. e



5y Ϫ 3x ϭ Ϫ5

3x ϩ 2y ϭ 19



41. e



2x ϭ Ϫ3y ϩ 17

4x Ϫ 5y ϭ 12



42. e



2y ϭ 5x ϩ 2

Ϫ4x ϭ 17 Ϫ 6y



43. e



0.5x ϩ 0.4y ϭ 0.2

0.2x ϩ 0.3y ϭ 0.8

44. e

0.3y ϭ 1.3 ϩ 0.2x

0.3x ϩ 0.4y ϭ 1.3



45. e



0.32m Ϫ 0.12n ϭ Ϫ1.44

Ϫ0.24m ϩ 0.08n ϭ 1.04



46. e



0.06g Ϫ 0.35h ϭ Ϫ0.67

Ϫ0.12g ϩ 0.25h ϭ 0.44



47. e



Ϫ16u ϩ 14v ϭ 4

1

2

2 u Ϫ 3 v ϭ Ϫ11



x ϩ 13y ϭ Ϫ2

1

2x ϩ 5y ϭ 3

3



48. e 43



Solve using any method and identify the system as

consistent, inconsistent, or dependent. Verify solutions

using a graphing calculator.



49. e



4x ϩ 34y ϭ 14

Ϫ9x ϩ 58y ϭ Ϫ13



2

xϩyϭ2

50. e 3

2y ϭ 56x Ϫ 9



51. e



0.2y ϭ 0.3x ϩ 4

1.2x ϩ 0.4y ϭ 5

52. e

0.6x Ϫ 0.4y ϭ Ϫ1

0.5y ϭ Ϫ1.5x ϩ 2



53. e



6x Ϫ 22 ϭ Ϫy

3x ϩ 12y ϭ 11



55. e



Ϫ10x ϩ 35y ϭ Ϫ5

2x ϩ 3y ϭ 4

56. e

y ϭ 0.25x

x ϭ Ϫ2.5y



57. e



7a ϩ b ϭ Ϫ25

2a Ϫ 5b ϭ 14



58. e



Ϫ2m ϩ 3n ϭ Ϫ1

5m Ϫ 6n ϭ 4



59. e



4a ϭ 2 Ϫ 3b

6b ϩ 2a ϭ 7



60. e



3p Ϫ 2q ϭ 4

9p ϩ 4q ϭ Ϫ3



54. e



15 Ϫ 5y ϭ Ϫ9x

Ϫ3x ϩ 53y ϭ 5



WORKING WITH FORMULAS



61. Uniform motion with current: e



1R ؉ C2T1 ‫ ؍‬D1

1R ؊ C2T2 ‫ ؍‬D2



The formula shown can be used to solve uniform motion problems involving a current, where D represents

distance traveled, R is the rate of the object with no current, C is the speed of the current, and T is the time.

Chan-Li rows 9 mi up river (against the current) in 3 hr. It only took him 1 hr to row 5 mi downstream (with the

current). How fast was the current? How fast can he row in still water?



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62. Fahrenheit and Celsius temperatures: e



y ‫ ؍‬95x ؉ 32

y ‫ ؍‬59 1x ؊ 322



؇F

؇C



Many people are familiar with temperature measurement in degrees Celsius and degrees Fahrenheit, but few realize

that the equations are linear and there is one temperature at which the two scales agree. Solve the system using the

method of your choice and find this temperature.





APPLICATIONS



Solve each application by modeling the situation with a

linear system. Be sure to clearly indicate what each

variable represents. Check answers using a graphing

calculator and the method of your choice.



Mixture

63. Theater productions: At a recent production of A

Comedy of Errors, the Community Theater brought

in a total of $30,495 in revenue. If adult tickets

were $9 and children’s tickets were $6.50, how

many tickets of each type were sold if 3800 tickets

in all were sold?

64. Milkfat requirements: A dietician needs to mix 10

gal of milk that is 212 % milkfat for the day’s rounds.

He has some milk that is 4% milkfat and some that

is 112 % milkfat. How much of each should be used?

65. Filling the family cars: Cherokee just filled both

of the family vehicles at a service station. The total

cost for 20 gal of regular unleaded and 17 gal of

premium unleaded was $144.89. The premium gas

was $0.10 more per gallon than the regular gas.

Find the price per gallon for each type of gasoline.

66. Household cleaners: As a cleaning agent, a solution

that is 24% vinegar is often used. How much pure

(100%) vinegar and 5% vinegar must be mixed to

obtain 50 oz of a 24% solution?

67. Alumni contributions: A wealthy alumnus

donated $10,000 to his alma mater. The college

used the funds to make a loan to a science major at

7% interest and a loan to a nursing student at 6%

interest. That year the college earned $635 in

interest. How much was loaned to each student?

68. Investing in bonds: A total of $12,000 is invested

in two municipal bonds, one paying 10.5% and the

other 12% simple interest. Last year the annual

interest earned on the two investments was $1335.

How much was invested at each rate?

69. Saving money: Bryan has been doing odd jobs

around the house, trying to earn enough money to buy

a new Dirt-Surfer©. He saves all quarters and dimes

in his piggy bank, while he places all nickels and

pennies in a drawer to spend. So far, he has 225 coins



in the piggy bank, worth a total of $45.00. How many

of the coins are quarters? How many are dimes?

70. Coin investments: In 1990, Molly attended a coin

auction and purchased some rare “Flowing Hair”

fifty-cent pieces, and a number of very rare twocent pieces from the Civil War Era. If she bought

47 coins with a face value of $10.06, how many of

each denomination did she buy?

Uniform Motion

71. Canoeing on a stream: On a recent camping trip,

it took Molly and Sharon 2 hr to row 4 mi upstream

from the drop in point to the campsite. After a

leisurely weekend of camping, fishing, and

relaxation, they rowed back downstream to the

drop in point in just 30 min. Use this information

to find (a) the speed of the current and (b) the

speed Sharon and Molly would be rowing in still

water.

72. Taking a luxury cruise: A luxury ship is taking a

Caribbean cruise from Caracas, Venezuela, to just off

the coast of Belize City on the Yucatan Peninsula, a

distance of 1435 mi. En route they encounter the

Caribbean Current, which flows to the northwest,

parallel to the coastline. From Caracas to the Belize

coast, the trip took 70 hr. After a few days of fun in

the sun, the ship leaves for Caracas, with the return

trip taking 82 hr. Use this information to find (a) the

speed of the Caribbean Current and (b) the cruising

speed of the ship.

73. Airport walkways: As part of an algebra field trip,

Jason takes his class to the airport to use their

moving walkways for a demonstration. The class

measures the longest walkway, which turns out to

be 256 ft long. Using a stop watch, Jason shows it

takes him just 32 sec to complete the walk going in

the same direction as the walkway. Walking in a

direction opposite the walkway, it takes him

320 sec—10 times as long! The next day in class,

Jason hands out a two-question quiz: (1) What was the

speed of the walkway in feet per second? (2) What is

my (Jason’s) normal walking speed? Create the

answer key for this quiz.



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74. Racing pigeons: The American Racing Pigeon

Union often sponsors opportunities for owners to

fly their birds in friendly competitions. During a

recent competition, Steve’s birds were liberated in

Topeka, Kansas, and headed almost due north to

their loft in Sioux Falls, South Dakota, a distance

of 308 mi. During the flight, they encountered a

steady wind from the north and the trip took 4.4 hr.

The next month, Steve took his birds to a

competition in Grand Forks, North Dakota, with

the birds heading almost due south to home, also a

distance of 308 mi. This time the birds were aided

by the same wind from the north, and the trip took

only 3.5 hr. Use this information to (a) find the

racing speed of Steve’s birds and (b) find the speed

of the wind.

75. Lawn service: Dave and his sons run a lawn

service, which includes mowing, edging, trimming,

and aerating lawns. His fixed cost includes

insurance, his salary, and monthly payments on

equipment, and amounts to $4000/mo. The variable

costs include gas, oil, hourly wages for his

employees, and miscellaneous expenses, which run

about $75 per lawn. The average charge for fullservice lawn care is $115 per visit. Do a breakeven analysis to (a) determine how many lawns

Dave must service each month to break even and

(b) the revenue required to break even.

76. Production of mini-microwave ovens: Due to

high market demand, a manufacturer decides to

introduce a new line of mini-microwave ovens for

personal and office use. By using existing factory

space and retraining some employees, fixed costs

are estimated at $8400/mo. The components to

assemble and test each microwave are expected to

run $45 per unit. If market research shows

consumers are willing to pay at least $69 for this

product, find (a) how many units must be made and

sold each month to break even and (b) the revenue

required to break even.

77. Farm commodities: One area where the law of

supply and demand is clearly at work is farm

commodities. Both growers and consumers watch

this relationship closely, and use data collected by

government agencies to track the relationship and

make adjustments, as when a farmer decides to

convert a large portion of her farmland from corn

to soybeans to improve profits. Suppose that for

x billion bushels of soybeans, supply is modeled by

y ϭ 1.5x ϩ 3, where y is the current market price

(in dollars per bushel). The related demand

equation might be y ϭ Ϫ2.20x ϩ 12. (a) How

many billion bushels will be supplied at a market

price of $5.40? What will the demand be at this



price? Is supply less than demand? (b) How many

billion bushels will be supplied at a market price

of $7.05? What will the demand be at this price?

Is demand less than supply? (c) To the nearest

cent, at what price does the market reach

equilibrium? How many bushels are being

supplied/demanded?

78. Digital media: Market

research has indicated

that by 2015, sales of

MP3 players and similar

products will mushroom

into a $70 billion dollar

market. With a market

this large, competition is

often fierce—with

suppliers fighting to

earn and hold market shares. For x million MP3

players sold, supply is modeled by y ϭ 10.5x ϩ 25,

where y is the current market price (in dollars).

The related demand equation might be

y ϭ Ϫ5.20x ϩ 140. (a) How many million MP3

players will be supplied at a market price of $88?

What will the demand be at this price? Is supply less

than demand? (b) How many million MP3 players

will be supplied at a market price of $114? What will

the demand be at this price? Is demand less than

supply? (c) To the nearest cent, at what price does the

market reach equilibrium? How many units are being

supplied/demanded?

79. Pricing wakeboards: A water sports company

that manufactures high-end wakeboards has hired

an outside consulting firm to do some market

research on their wakeboard. This consulting firm

collected the following supply and demand data for

this and comparable wakeboards over a 10-week

period. Find the equilibrium point. Round your

answer to the nearest integer and dollar.

Average Price

(in U.S. dollars)



Quantity

Demanded



Available

Inventory



424.85



175



232



445.25



166



247



389.55



291



215



349.98



391



201



402.22



218



226



413.87



200



222



481.73



139



251



419.45



177



235



397.05



220



219



361.90



317



212



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CHAPTER 6 Systems of Equations and Inequalities



80. Pricing pet care products: A metal shop that

manufactures pens for pet rabbits has collected

some data on sales and production over the past

8 weeks. The following table shows the supply and

demand data for these pens. Find the equilibrium

point (round to the nearest cent and whole cage).

Average Price

(in U.S. dollars)



Quantity Sold

(Demand)



Production

(Supply)



Average Price Quantity Demanded Available Inventory

(in U.S. dollars)

(in millions)

(in millions)

9.40



0.84



1.23



8.51



1.17



0.95



8.78



1.05



1.11



10.82



0.68



1.29



6.77



1.47



0.77



9.33



0.91



1.21



1.25



0.88



22.99



12



7



8.34



21.49



14



6



10.37



0.76



1.27



1.09



1.02



23.99



11



7



8.62



26.99



9



11



8.44



1.21



0.92



1.18



0.97



1.01



1.17



25.99



8



10



8.58



27.99



8



13



8.96



24.49



10



9



26.49



9



11



81. Tracking supply and demand— oil products:

The U.S. Bureau of Labor and Statistics tracks

important data from many different markets. In

May 2008, it collected the following supply-anddemand data for refined gasoline. Data were

collected every Tuesday and Friday. Find the

equilibrium point, rounding your answer to the

nearest hundred thousand gallons and whole cent.

Average Price Quantity Demanded Available Inventory

(in U.S. dollars)

(1 ؋ 107 gal)

(1 ؋ 107 gal)

3.17



8.82



9.10



3.12



8.87



9.05



3.04



9.08



8.97



2.84



9.22



8.91



3.11



8.92



9.02



3.15



8.76



9.08



3.10



9.01



8.99



3.11



8.94



9.01



2.93



9.13



8.93



82. Tracking supply and demand—energy efficient

lightbulbs: The U.S. Bureau of Labor and

Statistics has collected the following supply and

demand data for the energy-efficient fluorescent

lightbulbs sold each month for the past year. Find

the equilibrium point, rounding your answer to the

nearest ten thousand lightbulbs and whole cent.

What is the yearly demand at the equilibrium

point?



Descriptive Translation

83. Important dates in U.S. history: If you sum the

year that the Declaration of Independence was

signed and the year that the Civil War ended, you

get 3641. There are 89 yr that separate the two

events. What year was the Declaration signed?

What year did the Civil War end?

84. Architectural

wonders: When it

was first

constructed in

1889, the Eiffel

Tower in Paris,

France, was the

tallest structure in

the world. In

1975, the CN

Tower in Toronto, Canada, became the world’s tallest

structure. The CN Tower is 153 ft less than twice the

height of the Eiffel Tower, and the sum of their

heights is 2799 ft. How tall is each tower?

85. Pacific islands land area: In the South Pacific, the

island nations of Tahiti and Tonga have a combined

land area of 692 mi2. Tahiti’s land area is 112 mi2

more than Tonga’s. What is the land area of each

island group?

86. Card games: On a cold winter night, in the lobby

of a beautiful hotel in Sante Fe, New Mexico, Marc

and Klay just barely beat John and Steve in a close

game of Trumps. If the sum of the team scores was

990 points, and there was a 12-point margin of

victory, what was the final score?



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591



Given any two points, the equation of a line through these points can be found using a system of equations. While

there are certainly more efficient methods, using a system here will show how we can find equations for polynomials

of higher degree. The key is to note that each point will yield an equation of the form y ‫ ؍‬mx ؉ b. For instance, the

points (3, 6) and 1؊2, ؊42 yield the system e



6 ‫ ؍‬3m ؉ b

.

؊4 ‫ ؍‬؊2m ؉ b



87. Use a system of equations to find the equation of the line containing the points (2, 7) and 1Ϫ4, Ϫ52 .

88. Use a system of equations to find the equation of the line containing the points 19, Ϫ12 and 1Ϫ3, 72 .





EXTENDING THE CONCEPT



89. Federal income tax reform has been a hot political

topic for many years. Suppose tax plan A calls for

a flat tax of 20% tax on all income (no deductions

or loopholes). Tax plan B requires taxpayers to pay

$5000 plus 10% of all income. For what income

level do both plans require the same tax?





90. Suppose a certain amount of money was invested at

6% per year, and another amount at 8.5% per year,

with a total return of $1250. If the amounts

invested at each rate were switched, the yearly

income would have been $1375. To the nearest

whole dollar, how much was invested at each rate?



MAINTAINING YOUR SKILLS



91. (4.2) Use the rational zeroes theorem to write the

polynomial in completely factored form:

3x4 Ϫ 19x3 ϩ 15x2 ϩ 27x Ϫ 10.



92. (2.2) Given the tool box function f 1x2 ϭ ͿxͿ, sketch

the graph of F1x2 ϭ ϪͿx ϩ 3ͿϪ2.



6.2



93. (3.2) Graph y ϭ x2 Ϫ 6x Ϫ 16 and state the

interval where f 1x2 Յ 0.

94. (5.5) Solve for x (rounded to the nearest

thousandth): 33 ϭ 77.5eϪ0.0052x Ϫ 8.37.



Linear Systems in Three Variables with Applications



LEARNING OBJECTIVES

In Section 6.2 you will see

how we can:



A. Visualize a solution in



The transition to systems of three equations in three variables requires a fair amount of

“visual gymnastics” along with good organizational skills. Although the techniques

used are identical and similar results are obtained, the third equation and variable give

us more to track, and we must work more carefully toward the solution.



three dimensions



B. Check ordered triple

solutions

C. Solve linear systems in

three variables

D. Recognize inconsistent

and dependent systems

E. Use a system of three

equations in three

variables to solve

applications



A. Visualizing Solutions in Three Dimensions

The solution to an equation in one variable is the single number that satisfies the equation. For x ϩ 1 ϭ 3, the solution is x ϭ 2 and its graph is a single point on the number

line, a one-dimensional graph. The solution to an equation in two variables, such as

x ϩ y ϭ 3, is an ordered pair (x, y) that satisfies the equation. When we graph this

solution set, the result is a line on the xy-coordinate grid, a two-dimensional graph.

The solutions to an equation in three variables, such as x ϩ y ϩ z ϭ 6, are the ordered

triples (x, y, z) that satisfy the equation. When we graph this solution set, the result is a

plane in space, a graph in three dimensions. Recall a plane is a flat surface having infinite length and width, but no depth. We can graph this plane using the intercept

method and the result is shown in Figure 6.15. For graphs in three dimensions, the xyplane is parallel to the ground (the y-axis points to the right) and z is the vertical axis.

To find an additional point on this plane, we use any three numbers whose sum is 6,

such as (2, 3, 1). Move 2 units along the x-axis, 3 units parallel to the y-axis, and 1 unit

parallel to the z-axis, as shown in Figure 6.16.



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Figure 6.15



WORTHY OF NOTE

We can visualize the location of a

point in space by considering a

large rectangular box 2 ft long ϫ

3 ft wide ϫ 1 ft tall, placed snugly

in the corner of a room. The floor is

the xy-plane, one wall is the

xz-plane, and the other wall is the

yz-plane. The z-axis is formed

where the two walls meet and the

corner of the room is the origin

(0, 0, 0). To find the corner of the

box located at (2, 3, 1), first locate

the point (2, 3) in the xy-plane (the

floor), then move up 1 ft.



Figure 6.16



z



z



(0, 0, 6)



(0, 0, 6)



(2, 3, 1)

y



2 units along x



y







EXAMPLE 1



x



y



z



l

lle



x



(0, 6, 0)



l

lle



(6, 0, 0)



a ra

sp

nit

3u



(6, 0, 0)



a

par

nit

1u



(0, 6, 0)



Finding Solutions to an Equation in Three Variables

Use a guess-and-check method to find four additional points on the plane

determined by x ϩ y ϩ z ϭ 6.







Solution

A. You’ve just seen how

we can visualize a solution in

three dimensions



We can begin by letting x ϭ 0, then use any combination of y and z that sum to 6.

Two examples are (0, 2, 4) and (0, 5, 1). We could also select any two values for x

and y, then determine a value for z that results in a sum of 6. Two examples are

1Ϫ2, 9, Ϫ12 and 18, Ϫ3, 12.

Now try Exercises 7 through 10 ᮣ



B. Solutions to a System of Three Equations in Three Variables

When solving a system of three equations in three variables, remember each equation

represents a plane in space. These planes can intersect in various ways, creating different

possibilities for a solution set (see Figures 6.17 to 6.20). The system could have a unique

solution (a, b, c), if the planes intersect at a single point (Figure 6.17) (the point satisfies

all three equations simultaneously). If the planes intersect in a line (Figure 6.18), the system is linearly dependent and there is an infinite number of solutions. Unlike the twodimensional case, the equation of a line in three dimensions is somewhat complex, and the

coordinates of all points on this line are usually represented by a specialized ordered triple,

which we use to state the solution set. If the planes intersect at all points, the system has

coincident dependence (see Figure 6.18). This indicates the equations of the system

differ by only a constant multiple—they are all “disguised forms” of the same equation.

The solution set is any ordered triple (a, b, c) satisfying this equation. Finally, the system

may have no solutions. This can happen a number of different ways, most notably if

the planes either intersect or are parallel, as shown in Figure 6.20 (other possibilities

are discussed in the exercises). In the case of “no solutions,” an ordered triple may

satisfy none of the equations, only one of the equations, only two of the equations, but

not all three equations.

Figure 6.17



Unique solution



Figure 6.18



Linear

dependence



Figure 6.19



Coincident

dependence



Figure 6.20



No solutions



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