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
xϭ
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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Section 6.1 Linear Systems in Two Variables with Applications
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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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