Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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SOLUTION Any solution of the above system must also be a solution of this
system:
10x 6y 6
[First equation multiplied by 2]
9x 6y 3. [Second equation multiplied by 3]
The multipliers 2 and 3 were chosen so that the coefficients of y in the new
equations would be negatives of each other. Any solution of this last system must
also be a solution of the equation obtained by adding these two equations.
10x 6y 6
9x 6y 3
x 3. [The second variable has been eliminated]
Substituting x 3 in the first of the original equations shows that
5(3) 3y 3
3y 12
y 4.
Therefore, the solution of the original system is x 3, y 4. ■
EXAMPLE 5
To solve the system
2x 3y 5
4x 6y 1,
we multiply the first equation by 2 and add.
4x 6y 10
4x 6y 1
0 9.
Since 0 9 is always false, the original system cannot possibly have any solu-
tions. A system with no solution is said to be inconsistent. ■
EXAMPLE 6
To solve the system
3x y 2
6x 2y 4
776 CHAPTER 11 Systems of Equations
Confirm the result of Example 5 geometrically by graphing the two equations in the
system. Do these lines intersect, or are they parallel?
GRAPHING EXPLORATION
we multiply the first equation by 2 to obtain the following system.
6x 2y 4
6x 2y 4.
The two equations are identical. So the solutions of this system are the same as the
solutions of the single equation 6x 2y 4, which can be rewritten as
2y 6x 4
y 3x 2.
This equation, and hence the original system, has infinitely many solutions. They
can be described as follows: Choose any real number for x, say, x b. Then
y 3x 2 3b 2. So the solutions of the system are all pairs of numbers of
the form
x b, y 3b 2, where b is any real number.
A system such as this is said to be dependent. ■
Some nonlinear systems can be solved by replacing them with equivalent
linear systems.
EXAMPLE 7
Solve the following system:
1
x
3
y
1
2
x
1
y
5.
SOLUTION We wish to transform this system into a linear one. We let u 1/x
and v 1/y so that the system becomes
u 3v 1
2u v 5.
We can solve this system by multiplying the first equation by 2 and adding it to
the second equation.
2u 6v 2
2u v 5
7v 7
v 1.
Substituting v 1 in the equation u 3v 1, we see that
u 3v 1 3(1) 1 2.
Consequently, the possible solution of the original system is
x
1
u
1
2
and y
1
v
(
1
1)
1.
You should substitute this possible solution in both equations of the original sys-
tem to check that it is actually a solution of the system. ■
SECTION 11.1 Systems of Linear Equations in Two Variables 777
APPLICATIONS
Producers are happy to supply items at a high price, but if they do, the consumers’
demand for them may be low. At lower prices, more items will be demanded, but
if the price is too low, producers may not want to supply the items. The equilib-
rium price is the price at which the number of items demanded by consumers is
the same as the number supplied by producers. The number of items demanded
and supplied at the equilibrium price is the equilibrium quantity.
EXAMPLE 8
The consumer demand for a certain type of floor tile is related to its price by the
equation p 60 .75x, where p is the price (in dollars) at which x thousand boxes
of tile will be demanded. The supply of these tiles is related to their price by the
equation p .8x 5.75, where p is the price at which x thousand boxes will be sup-
plied by the producer. Find the equilibrium quantity and the equilibrium price.
SOLUTION We must find the values of x and p that satisfy both the supply and
demand equations. In other words, we must solve this system:
p 60 .75x
p .8x 5.75.
Algebraic Method. Since both equations are already solved for p, we use substitu-
tion. Substituting the value of p given by the first equation into the second, we obtain
60 .75x .8x 5.75
Subtract .8x from both sides: 60 1.55x 5.75
Subtract 60 from both sides: 1.55x 54.25
Divide both sides by 1.55: x
5
1
4
.
.
5
2
5
5
35.
We can determine p by substituting x 35 in either equation, say, the first one.
p 60 .75x
p 60 .75(35) 33.75.
Therefore, the equilibrium quantity is 35,000 boxes (x is measured in thousands),
and the equilibrium price is $33.75 per box.
Graphical Method. We graph the two equations in the form y 60 .75x and
y .8x 5.75 on the same screen and find their intersection point (Figure 11–4).
The intersection point (35, 33.75) is called the equilibrium point. Its first coordinate
is the equilibrium quantity, and its second coordinate is the equilibrium price. ■
EXAMPLE 9
575 people attend a ball game, and total ticket sales are $2575. If adult tickets cost
$5 and children’s tickets cost $3, how many adults attended the game? How many
children?
778 CHAPTER 11 Systems of Equations
80
−20
90
0
Figure 11–4
SOLUTION Let x be the number of adults, and let y be the number of children.
Then,
Number of adults Number of children Total attendance
x y 575.
We can obtain a second equation by using the information about ticket sales.
Adult ticket sales Children ticket sales Total ticket sales
644447444448 6444474444448
Price Number Price Number
per
of
per
of
2575
ticket adults ticket children
5x 3y 2575.
To find x and y, we need only solve this system of equations:
x y 575
5x 3y 2575.
Multiplying the first equation by 3 and adding, we have
3x 3y 1725
5x 3y 2575
2x 850
x 425.
So 425 adults attended the game. The number of children was
y 575 x 575 425 150. ■
EXAMPLE 10
A plane flies 3000 miles from San Francisco to Boston in 5 hours, with a tailwind
all the way. The return trip on the same route, now with a headwind, takes 6 hours.
Assuming that both remain constant, find the speed of the plane and the speed of
the wind.
SOLUTION Let x be the plane’s speed, and let y be the wind speed (both in
miles per hour). Then on the trip to Boston with a tailwind,
x y actual speed of the plane (wind and plane go in same direction).
On the return trip against a headwind,
x y actual speed of plane (wind and plane go in opposite directions).
SECTION 11.1 Systems of Linear Equations in Two Variables 779
—
———
———
————
————
—
—
Using the basic rate/distance equation, we have
Trip to Boston Return Trip
time rate distance time rate distance
5(x y) 3000 6(x y) 3000
5x 5y 3000 6x 6y 3000
x y 600 x y 500
Thus, we need only solve this system of equations:
x y 600
x y 500.
Adding the two equations shows that
2x 1100
x 550.
Substituting this result in the first equation, we have
550 y 600
y 50.
Thus, the plane’s speed is 550 mph, and the wind speed is 50 mph. ■
EXAMPLE 11
How many pounds of tin and how many pounds of copper should be added to
1000 pounds of an alloy that is 10% tin and 30% copper to produce a new alloy
that is 27.5% tin and 35% copper?
SOLUTION Let x be the number of pounds of tin and y the number of pounds
of copper to be added to the 1000 pounds of the old alloy. Then there will be
1000 x y pounds of the new alloy. We first find the amounts of tin and cop-
per in the new alloy.
Now consider the percentages of tin and copper in the new alloy.
780 CHAPTER 11 Systems of Equations
—
—
—
—
—
—
Pounds in Pounds in
Pounds added
old alloy new alloy
Tin 10% of 1000 x 100 x
Copper 30% of 1000 y 300 y
Percentage in Total weight
Pounds in new alloy
new alloy of new alloy
Tin 27.5% of 1000 x y .275(1000 x y)
Copper 35% of 1000 x y .35(1000 x y)
The two ways of computing the weight of each metal in the alloy must produce
the same result, that is,
100 x .275(1000 x y)
[pounds of tin]
300 y .35(1000 x y). [pounds of copper]
Multiplying out the right sides and rearranging terms produces this system of
equations:
.725x .275y 175
.35x .65y 50.
Multiplying the first equation by .65 and the second by .275 and adding the
results, we have
.47125x .17875y 113.75
.09625x .17875y 13.75
.37500x 127.50
x 340.
Substituting this in the first equation and solving for y shows that y 260. There-
fore, 340 pounds of tin and 260 pounds of copper should be added. ■
SECTION 11.1 Systems of Linear Equations in Two Variables 781
EXERCISES 11.1
In Exercises 1–6, determine whether the given values of x, y,
and z are a solution of the system of equations.
1. x 1, y 3 2. x 2, y 5
2x y 12x 3y 11
3x 2y 9 x 2y 12
3. x 2, y 1 4. x .3, y .7
1
3
x
1
2
y
1
6
4x 1.2y .36
3.1x 2y 4.7
1
2
x
1
3
y
2
3
5. x
1
3
, y 2, z 1 6. x 3, y
5
4
, z
3
4
3x y 2z 14x 6y 10z 27
4y 3z 54x 8y 28z 19
3z 3 x 4y 8z 8
In Exercises 7–14, use substitution to solve the system.
7. x 3y 6 8. 2x 3y 6
3x y 25x 7y 2
9. 3x 2y 4 10. 5x 3y 1
2x y 15x 2y 7
11. r s 0 12. t 7u 4
r s 5 t 5u 6
13. x y c d (where c, d are constants)
x y 2c d
14. x 2y 2c 5d (where c, d are constants)
3x y c 2d
In Exercises 15–30, use the elimination method to solve the
system.
15. 2x 2y 12 16. 4x 3y 10
2x 3y 10 7x 4y 1
17. x 3y 1 18. 3x 2y 12
2x y 5 x 5y 13
19. 2x 3y 15 20. 2x 3y 7
8x 12y 40 6x 9y 1
21. 4x 5y 2 22. 5x 2y 1
12x 15y 615x 6y 3
23.
3
x
2
y
3 24.
5
7
x
3
7
y
1
2
5
x
5
y
2 x
2
3
y
1
782 CHAPTER 11 Systems of Equations
25.
x
4
y
x
3
y
1 26.
x
3
y
x
5
y
2
3
x
4
y
x
2
y
9
x
4
3y
2x
5
y
3
8
27.
1
x
3
y
2 28.
3
x
2
y
7
2
x
1
y
3
1
x
4
y
14
29.
2
x
3
y
8 30.
x
3
7
y
2
4
7
3
x
1
y
1
x
5
7
y
1
4
3
Hint: Let u
x
1
7
and
v
y
1
4
.
31. The population y in year x of Philadelphia and Houston is
approximated by these equations:
Philadelphia: 7.96x y 1588.47
Houston: 26.67x y 1644.64,
where x 0 corresponds to 1990 and y is in thousands.*
(a) How can you tell from the equation whether a city’s
population was increasing or decreasing since 1990?
(b) According to this model, in what year did the two cities
have the same population?
32. The cost of bread, y, in dollars per pound, is approximated
by these equations:*
White Bread: .0071t y .982
Whole Wheat: .014t y 1.360,
where t is the number of years since 2000. According to this
model, in what year will white bread and wheat bread cost
the same?
33. On the basis of data from 2000–2004, the median income y
in year x for men and women is approximated by these
equations:
Men: 135x y 31,065
Women: 29.31x 3y 42,908
where x 0 corresponds to 2000 and y is in constant 2004
dollars.* If the equations remain valid in the future, when
will the median income of men and women be the same?
34. The death rate per 100,000 population y in year x for heart
disease and cancer is approximated by these equations:
Heart Disease: 6.9x 2y 728.4
Cancer: 1.3x y 167.5,
where x 0 corresponds to 1970.* If the equations remain
accurate, when will the death rates for heart disease and
cancer be the same?
In Exercises 35 and 36, solve the system of equations.
35. ax by r (where a, b, c, d, r, s are
cx dy s constants and ad bc 0)
36. ax by ab (where a, b are nonzero constants)
bx ay ab
37. Let c be any real number. Show that this system has exactly
one solution.
x 2y c
6x 3y 4.
38. (a) Find the values of c for which this system has an infi-
nite number of solutions.
2x 4y 6
3x 6y c.
(b) Find the values of c for which the system in part (a) has
no solutions.
In Exercises 39 and 40, find the values of c and d for which
both given points lie on the given straight line.
39. cx dy 2; (0, 4) and (2, 16)
40. cx dy 6; (1, 3) and (2, 12)
In Exercises 41–44, find the equilibrium quantity and the equi-
librium price. In the supply and demand equations, p is price
(in dollars) and x is quantity (in thousands).
41. Supply: p .85x
Demand: p 40 1.15x
42. Supply: p 1.4x .6
Demand: p 2x 3.2
43. Supply: p 300 30x
Demand: p 80 25x
44. Supply: p 181 .01x
Demand: p 52 .02x
45. A 200-seat theater charges $8 for adults and $5 for children.
If all seats were filled and the total ticket income was
$1,435, how many adults and how many children were in
the audience?
46. A theater charges $50 for main floor seats and $15 for bal-
cony seats. If all seats are sold, the ticket income is $27,000.
At one show, 25% of the main floor seats and 40% of the
balcony seats were sold, and ticket income was $8,550.
How many seats are on the main floor and how many are in
the balcony?
*U.S. Department of Health and Human Services.
*U.S. Census Bureau
SECTION 11.1 Systems of Linear Equations in Two Variables 783
47. A boat made a 4-mile trip upstream against a constant cur-
rent in 15 minutes. The return trip at the same constant
speed with the same current took 12 minutes. What is the
speed of the boat and what is the speed of the current?
48. A plane flying into a headwind travels 2000 miles in
4 hours and 24 minutes. The return flight along the same
route with a tailwind takes 4 hours. Find the wind speed and
the plane’s speed (assuming that both are constant).
49. At a certain store, cashews cost $4.40/pound and peanuts
cost $1.20/pound. If you want to buy exactly 6 pounds of
nuts for $12 dollars, how many pounds of each kind of nuts
should you buy?
50. How many cubic centimeters (cm
3
) of a solution that is 20%
acid and of another solution that is 45% acid should be mixed
to produce 100 cm
3
of a solution that is 30% acid?
51. How many grams of a 50%-silver alloy should be mixed with
a 75%-silver alloy to obtain 40 grams of a 60%-silver alloy?
52. A winemaker has two large casks of wine. One wine is 8%
alcohol, and the other is 18% alcohol. How many liters of
each wine should be mixed to produce 30 liters of wine that
is 12% alcohol?
53. Bill and Ann plan to install a heating system for their swim-
ming pool. Since gas is not available, they have a choice of
electric or solar heat. They have gathered the following cost
information.
(a) Ignoring changes in fuel prices, write a linear equation
for each heating system that expresses its total cost y in
terms of the number of years x of operation.
(b) What is the five-year total cost of electric heat? Of solar
heat?
(c) In what year will the total cost of the two heating sys-
tems be the same? Which is the cheaper system before
that time? After that time?
54. A toy company makes Boomie Babies, as well as collector
cases for each Boomie Baby. To make x cases cost the com-
pany $5000 in fixed overhead, plus $7.50 per case. An out-
side supplier has offered to produce any desired volume of
cases for $8.20 per case.
(a) Write an equation that expresses the company’s cost to
make x cases itself.
(b) Write an equation that expresses the cost of buying x
cases from the outside supplier.
(c) Graph both equations on the same axes and determine
when the two costs are the same.
(d) When should the company make the cases themselves
and when should they buy them from the outside sup-
plier?
Installation Monthly
System Costs Operational Cost
Electric $2,000 $80
Solar $14,000 $9.50
On Broadway Off Broadway
Initial cost to open
†
$1,295,000 $440,000
Weekly costs
†
$206,500 $82,000
Weekly revenue $250,500 $109,000
*Albert Goetz, “Basic Economics: Calculating against Theatrical
Disaster,” Mathematics Teacher 89, no. 1 (January 1996) © 1996.
Reprinted with permission of the National Councils of Teachers of
Mathematics. All rights reserved.
†
Initial cost includes sets, costumes, rehearsals, etc. Weekly expenses
include theater rent, salaries, advertising, etc.
55.
When Neil Simon planned to open his play London Suite,
his producer, Emanuel Azenberg, made the following cost
and revenue estimates for opening on Broadway or off
Broadway.*
For a production on Broadway that runs x weeks, find a lin-
ear equation that gives the
(a) total revenue R;
(b) total cost C;
(c) total profit P.
(d) Do parts (a)–(c) for an off-Broadway production.
(e) After how many weeks will the on-Broadway profit
equal the off-Broadway profit? What is the profit
then?
(f) When would it be better to open off Broadway rather
than on Broadway?
56. A store sells their 80-GB iPods for $350, and their 30-GB
iPods for $250. Their total iPod inventory would sell for
$15,250. During a recent month, the store actually sold half
of the 80-GB iPods and two-thirds of their 30-GB iPods,
taking in a total of $9,000. How many of each kind did they
have at the beginning of the month?
Exercises 57–60 deal with the break-even point, which is the
point at which revenue equals cost.
57. The cost C of making x hedge trimmers is given by
C 45x 6000. Each hedge trimmer can be sold for $60.
(a) Find an equation that expresses the revenue R from sell-
ing x hedge trimmers.
(b) How many hedge trimmers must be sold for the com-
pany to break even?
58. The cost C of making x cases of pet food is given by
C 100x 2600. Each case of 100 boxes sells for $120.
(a) Find an equation that expresses the revenue from sell-
ing x cases.
(b) How many cases must be sold for the company to break
even?
59. A firm is developing a new product. The marketing depart-
ment estimates that no more than 450 units can be sold.
Costs for producing x units are expected to be given by C
80x 7500, and revenues are given by R 95x. You are
the manager who must determine whether or not to produce
the product. What is your decision? Why?
60. Do Exercise 59 when costs are given by
C 1750x 96,175,
and revenue is given by R 1975x; no more than
800 units can be produced.
61. An investor has part of her money in an account that pays
9% annual interest and the rest in an account that pays 11%
annual interest. If she has $8000 less in the higher-paying
account than in the lower-paying one and her total annual
interest income is $2010, how much does she have invested
in each account?
62. Joyce has money in two investment funds. Last year, the
first fund paid a dividend of 8%, the second paid a dividend
of 2%, and Joyce received a total of $780. This year, the
first fund paid a 10% dividend, the second paid only 1%,
and Joyce received $810. How much money does she have
invested in each fund?
63. The table shows the percentages of men and women in the
labor force who are college graduates in selected years.*
(a) Use linear regression (with x 0 corresponding to 1990)
to find an equation that gives the percentage of men in
the labor force in year x who are college graduates.
784 CHAPTER 11 Systems of Equations
Year 1992 1995 1997 1998 1999 2000 2005 2006
Men 27.5 29.7 29.2 29.6 30.5 30.9 32.1 32.5
Women 25.0 26.6 28.0 28.6 29.5 29.8 32.8 33.5
*U.S. Bureau of Labor Statistics.
(b) Do part (a) for women.
(c) When did the percentage of women equal that of men?
64. The table shows the number of passenger cars (in thou-
sands) imported into the United States from Japan and
Canada in selected years.*
(a) Use linear regression to find an equation that approxi-
mates the number y of cars imported from Japan in year
x, with x 5 corresponding to 1995.
(b) Do part (a) for Canada.
(c) If these models remain accurate, in what year will the
imports from Japan and Canada be the same? Approxi-
mately how many cars will be imported that year?
65. A machine in a pottery factory takes 3 minutes to form a
bowl and 2 minutes to form a plate. The material for a bowl
costs $.25, and the material for a plate costs $.20. If the ma-
chine runs for 8 hours straight and exactly $44 is spent for
material, how many bowls and plates can be produced?
66. Because Chevrolet and Saturn produce cars in the same
price range, Chevrolet’s sales are a function not only of
Chevy prices (x), but of Saturn prices (y) as well. Saturn
prices are related similarly to both Saturn and Chevy prices.
Suppose General Motors forecasts the demand z
1
for
Chevrolets and the demand z
2
for Saturns to be given by
z
1
68,000 6x 4y and z
2
42,000 3x 3y.
Solve this system of equations and express
(a) the price x of Chevrolets as a function of z
1
and z
2
;
(b) the price y of Saturns as a function of z
1
and z
2
.
Year 1995 1996 1997 1998 1999 2000
Japan 1114.4 1190.9 1387.8 1456.1 1707.3 1839.1
Canada 1552.7 1690.7 1731.2 1837.6 2170.4 2138.8
*U.S. Census Bureau, Foreign Trade Division.
■ Solve systems of nonlinear equations algebraically and graphically.
■ Use systems of nonlinear equations to solve applied problems.
Some systems that include nonlinear equations can be solved algebraically.
EXAMPLE 1
Solve the system
2x y 1
xy 3.
11.1.A SPECIAL TOPICS Systems of Nonlinear Equations
Section Objectives
SOLUTION Solve the first equation for y:
y 2x 1
and substitute this into the second equation:
xy 3
x(2x 1) 3
2x
2
x 3
2x
2
x 3 0
(2x 3)(x 1) 0
2x 3 0orx 1 0
x 3/2 x 1.
Using the equation y 2x 1 to find the corresponding values of y, we see that
If x
3
2
, then y 2
3
2
1 2.
If x 1, then y 2(1) 1 3.
Therefore, the solutions of the system are x 3/2, y 2, and x 1, y 3.
■
EXAMPLE 2
Solve the system
x
2
y
2
8
x
2
y 6.
SOLUTION Solve the second equation for y, obtaining y x
2
6, and sub-
stitute this into the first equation.
x
2
y
2
8
x
2
(x
2
6)
2
8
x
2
x
4
12x
2
36 8
x
4
11x
2
28 0
(x
2
4)(x
2
7) 0
x
2
4 0orx
2
7 0
x
2
4 x
2
7
x 2 x 7
.
Using the equation y x
2
6 to find the corresponding values of y, we find that
the solutions of the system are
x 2, y 2; x 2, y 2; x 7
, y 1;
x 7
, y 1. ■
SPECIAL TOPICS 11.1.A Systems of Nonlinear Equations 785