Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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We must solve the resulting system.
49a 7b c 9
2209a 47b c 38
18,496a 136b c 70
whose matrix form is
49 7 1 a 9
2209 47 1
b
38
.
18,496 136 1 c 70
A calculator shows that the solution is
a 49 7 1 9 .00283
b
2209 47 1
1
38
.87798
.
c 18,496 136 1 70 2.99296
Therefore, the approximate equation for the ball’s path is
y .00283x
2
.87798x 2.99296.
(b) The ball hits the ground at a point where its height y is 0, that is, when
.00283x
2
.87798x 2.99296 0.
Use the quadratic formula or an equation solver to verify that the solutions of
this equation are x 3.37 (which is not applicable here) and x 313.61.
Therefore, the ball hits the ground approximately 314 feet from Matt. ■
Given any three points not on a straight line, the method in Examples 8 and 9
can be used to find the unique parabola that passes through these points. This
parabola can also be found by quadratic regression.*
To see why regression produces the parabola that actually passes through the
three points, recall that the error in a quadratic model is measured by taking the dif-
ference between the y-coordinate of each data point and the y-coordinate of the cor-
responding point on the model and summing the squares of these errors. If the data
points actually lie on a parabola (which is always the case with three points that are
not on a line), then the error for that parabola will be 0 (the smallest possible error).
Hence, it will be the parabola produced by the least squares quadratic regression
procedure.
816 CHAPTER 11 Systems of Equations
EXERCISES 11.3
In Exercises 1–6, determine whether the product AB or BA is
defined. If a product is defined, state its size (number of rows
and columns). Do not actually calculate any products.
1.
,
2. A
, B
1
4
7
3
8
0
9
3
0
5
5
0
0
0
0
2591
B
7006
1387
367
A
801
10
3. A
11
,
01
4. A
, B
415
5. A
3 7
,
210
0
7
4
1
0
8
1
6
CALCULATOR EXPLORATION
Use quadratic regression on the
three points in Example 9 and
verify that the equation obtained
is the same as the one in
Example 9.
*If you have not read Section 2.5 and Special Topics 4.4.A, you may skip this discussion.
5611
B
7815
12
B
34
SECTION 11.3 Matrix Methods for Square Systems 817
6. A (1 5 6), B
In Exercises 7–12, find AB.
7. A
, B
8. A
, B
9. A
, B
10. A
, B
201
11 2
11. A
,
023
23 0
12. A
, B
In Exercises 13–16, show that AB is not equal to BA by com-
puting both products.
13. A
, B
14. A
, B
421175
15. A
01 2
, B
2 26
301 000
16. A
, B
In Exercises 17–24, find the inverse of the matrix, if it exists.
12 35
17.
18.
34 12
19.
20.
4
3
4
1
1
3
1
0
2
4
6
2
3
0
1
2
5
2
4
7
0
2
7
5
4
4
1
0
1
4
6
3
5
2
5
6
2
3
0
0
3
1
8
1
2
5/4
3
9/4
2
2
1
5/4
3/4
1
2
1
3
3
5
2
1
8
3
0
4
1
11
6
1
8
2
3
4
1
0
1
2
1
5
6
4
0
2
8
7
7
2
0
1
2
5
7
7
3
1
6
1
5
0
3
1
0
1
1
1
0
2
1
0
0
3
4
1
0
2
0
1
1
3
5
1
2
4
0
9
3
0
5
3
7
1
2
1
3
0
2
1
1
7
2
3
1
4
7
8
1
12 0 131
21.
3 12
22.
213
23213 2
50 2 251
23.
22 1
24.
4 70
311817 2
In Exercises 25–28, solve the system of equations by using the
method of Example 6.
25. x y 1 26. 8x 3y 4z 47
x z 24x 5y 33
6x 2y 3z 3 x 2y 2z 0
27. 2x y 0 28. 7x 3y z 24
4x y 3z 12x z 2
3x y 2z 23x 2y z 21
In Exercises 29–39, solve the system by any method.
29. x y 2w 3
2x y z w 5
3x 3y 2z 2w 0
x 2y z 2
30. x 7y z 5w 70
2x y 8z 6
3y 7z w 20
3x 2y 5z 11
31. x y 6z 2v 1.5
x 5z 2v 3w 2
3x 2y 17z 6v 4w 2.5
4x 3y 21z 7v 2w 3
6x 5y 36z 12v 3w 3.5
32.
33.
x 2y 2z 2w 23
4x 4y z 5w 7
2x 5y 6z 4w 0
5x 13y 7z 12w 7
34. 2x 2y z 5w 0
3x y 5z 8w 1
y 3z w 3
x 2y 5z 4w 2
1
20
51
20
9
w
2w
3.5w
w
2.5v
3v
1.5v
4v
4.5z
2z
8.5z
6z
3.5z
1.5y
2.5y
y
y
x
x
2x
x
1011
B
1101
1110
818 CHAPTER 11 Systems of Equations
35. x 2y 5z 2v 4w 0
2x 4y 6z v 4w 0
5x 2y 3z 2v 3w 0
6x 5y 2z 5v 3w 0
x 2y z 2v 4w 0
36. 2x y 2z u v 3w 1
3x y z u 2v w 2
x y 3z u 5v w 5
5x y 4z u 2v 2w 1
x y z 3u 7v w 3
x y 3u 4w 0
37. x 2y 3z 1
3x 2y 4z 1
2x 6y 8z w 3
2x 2z 2w 3
38. 2x 6y z 4
5x z 4
4x 18y z 4
39. x 3w 2
x 4y z 3w 7
4y z 5
x 12y 3z 3w 17
In Exercises 40–42, find constants a, b, c such that the three
given points lie on the parabola y ax
2
bx c. See
Example 8.
40. (3, 2), (1, 1), (2, 1)
41. (3, 15), (1, 7), (5, 111)
42. (1, 2), (3, 1), (4, 1)
43. Concentrations of the greenhouse gas carbon dioxide (CO
2
)
have increased quadratically over the past half-century.* So
the concentration y of CO
2
(in parts per million) in year x is
given by an equation of the form
y ax
2
bx c.
(a) Let x 0 correspond to 1958 and use the following
data to find a, b, and c.
(b) Use this equation to estimate the CO
2
concentration in
the years 1983, 1993, and 2003. [For comparison pur-
poses, the actual concentrations in 1983 and 1993 were
343 ppm and 357 ppm respectively.]
44. The table shows the per capita consumption of chicken (in
pounds) in selected years.*
(a) Let x 0 correspond to 1990. Find a quadratic equation
that models this data.
(b) Use the equation to estimate the consumption of
chicken in 1995 and 2015. [The actual consumption in
1995 was 49.0 pounds.]
45. The table shows the per capita sales (in dollars) of electron-
ics and appliances in selected years.
†
(a) Find a quadratic equation that models this data, with
x 2 corresponding to 1992.
(b) Use the equation to estimate sales of electronics and
appliances in 1993, 1998, and 2002.
46. The table shows the percentage y of total consumer spending
on sound recordings that is spent on pop music in year x.
‡
(a) Assume that the data can be modeled by an equation of
the form
y ax
4
bx
3
cx
2
dx k.
(Let x 0 correspond to 1998.) Use the same method
used in Exercises 43–45 (with five equations instead of
three) to find the constants a, b, c, d, and k.
(b) Use the equation in part (a) to estimate the percentage
of consumer spending spent on pop music in 2005.
47. Find constants a, b, c such that the points (0, 2), (ln 2, 1),
and (ln 4, 4) lie on the graph of f (x) ae
x
be
x
c.
[Hint: Proceed as in Example 8.]
48. Find constants a, b, c such that the points (0, 1), (ln 2, 4),
and (ln 3, 7) lie on the graph of f(x) ae
x
be
x
c.
49. New Army Stores, a national chain of casual clothing stores,
recently sent shipments of jeans, jackets, sweaters, and
shirts to its stores in various cities. The number of items
shipped to each city and their total wholesale cost are shown
in the table on the next page. Find the wholesale price of
each of the following: one pair of jeans, one jacket, one
sweater, and one shirt.
Year 1958 1979 2001
CO
2
Concentration 315 337 371
*C. D. Keeling and T. P. Whorf, Scripps Institution of Oceanography.
*U.S. Department of Agriculture.
†
U.S. Census Bureau.
‡
Recording Industry Association of America, Inc.
Year 1992 1996 2000
Sales $169 $260 $320
Year 1998 2000 2002 2004 2006
% Pop 10 11 9 10 7.1
Year 1990 1999 2004
Chicken Consumption 42.4 54.2 59.2
CHAPTER 11 Review 819
50. A 100-bed nursing home provides two levels of long-term
care: regular and maximum. Patients at each level have a
choice of a private room or a less expensive semiprivate
room. The table below shows the number of patients in each
category at various times last year and the total daily cost for
these patients. Find the daily cost of each of the following: a
private room (regular care), a private room (maximum care),
a semiprivate room (regular care), and a semiprivate room
(maximum care).
51. A candy company produces three types of gift boxes:
A, B, and C. A box of variety A contains .6 pound of
chocolates and .4 pound of mints. A box of variety B
contains .3 pound of chocolates, .4 pound of mints, and
.3 pound of caramels. A box of variety C contains
.5 pound of chocolates, .3 pound of mints, and .2 pound
of caramels. The company has 41,400 pounds of
chocolates, 29,400 pounds of mints, and 16,200 pounds
of caramels in stock. How many boxes of each variety
should be made to use up all the stock?
52. Certain circus animals are fed the same three food
mixes: R, S, and T. Lions receive 1.1 units of R, 2.4
units of S, and 3.7 units of T each day. Horses receive
8.1 units of R, 2.9 units of S, and 5.1 units of T each
day. Bears receive 1.3 units of R, 1.3 units of S, and 2.3
units of T each day. If 16,000 units of R, 28,000 units
of S, and 44,000 units of T are available each day, how
many of each type of animal can be supported?
Total
City Jeans Jackets Sweaters Shirts Cost
Cleveland 3000 3000 2200 4200 $507,650
St. Louis 2700 2500 2100 4300 459,075
Seattle 5000 2000 1400 7500 541,225
Phoenix 7000 1800 600 8000 571,500
Regular Maximum
Care Patients Care Patients
Semi- Semi- Daily
Month private Private private Private Cost
January 22 8 60 10 $18,824
April 26 8 54 12 18,738
July 24 14 56 6 18,606
October 20 10 62 8 18,824
Chapter 11 Review
IMPORTANT CONCEPTS
Section 11.1
System of equations 772
Solution of a system 772
Number of solutions of a
system 773
Substitution method 774
Elimination method 775
Inconsistent system 776
Dependent system 777
Special Topics 11.1.A
Algebraic solution of systems of
nonlinear equations 784–785
Graphical solution of systems of
nonlinear equations 786–788
Section 11.2
Equivalent systems 792
Elementary operations 793
Back substitution 794
Triangular form system 794
Gaussian elimination 795
Augmented matrix 795
Row operations 795–796
Row echelon form matrix 797
Inconsistent system 798
Gauss-Jordan method 798
Reduced row echelon form matrix 799
Homogeneous system 800
Dependent system 800
Partial fraction decomposition 801
Number of solutions of a system
802
Section 11.3
m n matrix 807
Rows and columns 807
Equality of matrices 807
Multiplication of matrices 807–808
Identity matrix 810
Invertible matrix 810
Inverse of a matrix 810
Matrix form of a system of
equations 812
Inverses and solutions of a
system 813
820 CHAPTER 11 Systems of Equations
In Questions 1–4, solve the system of linear equations by any
means you want.
1. 5x 3y 4 2. 3x y 6
2x y 32x 3y 7
3. 3x 4y 7 4.
1
4
x
1
3
y
1
4
2x 3y 1
1
1
0
x
2
5
y
2
5
5. The number of days y in year x in which acceptable air qual-
ity standards were not met in two metropolitan areas are
approximated by these equations (in which x 4 corre-
sponds to 1994).*
Riverside–San Bernadino, CA: 10.89x y 184.92
Atlanta, GA: 16.06x 2y 32.04
In what year is the number of unacceptable days the same in
both areas? How many such days are there in that year?
6. The sum of one number and three times a second number is
20. The sum of the second number and two times the first
number is 55. Find the two numbers.
7. An alloy containing 40% gold and an alloy containing 80%
gold are to be mixed to produce 50 pounds of an alloy con-
taining 75% gold. How much of each alloy is needed?
8. The table shows the population (in thousands) of the cities
of Miami and Cleveland in selected years.
†
(a) Use linear regression to find an equation that gives the
population y of Miami in year x, with x 0 correspon-
ding to 1950.
(b) Do part (a) for Cleveland.
(c) If these models remain accurate, when do the two
cities have the same population and what is that popu-
lation?
In Questions 9–12, solve the nonlinear system.
9. x
2
y 0 10. x
2
2y
2
29
y 2x 3 x
2
5y 17
11. x
2
y
2
16 12. 2x
2
3xy y
2
63
x y 2 x
2
2xy 4y
2
124
13. Find the augmented matrix corresponding to this system.
3x 5y 2z 9
3x 2y 0
x y 3z 19
14. Use matrix methods to solve the system in Question 13.
15. Find the augmented matrix corresponding to this system.
x 7y z 2w 13
3x 5y z w 15
2x 2y 3z 19
16. Use matrix methods to solve the system in Question 15.
In Questions 17–22, solve the system.
17. 2x 5y z 8 18. x 3y 2z 6
x 2z 92x 3y 7z 1
2x 5y 2z 13 4x 3y 3z 11
19. 4x 3y 3z 2 20. 2x 6y z 4
5x 3y 2z 10 2x y 3z 0
2x 2y 3z 14 x 2y 1
21. x 2y 3z 1 22. 2x
1
2
y z 1
5y 10z 03x 4y z 8
8x 6y 4z 8 x 5y z 3
23. Let L be the line with equation 4x 2y 6, and let M be the
line with equation 10x 5y 15. Which of the fol-
lowing statements is true?
(a) L and M do not intersect.
(b) L and M intersect at a single point.
(c) L and M are the same line.
(d) All of the above are true.
(e) None of the above are true.
24. Which of the following statements about this system of
equations are false?
x z 2
6x 4y 14z 24
2x y 4z 7
(a) x 2, y 3, z 0 is a solution.
(b) x 1, y 1, z 1 is a solution.
(c) x 1, y 3, z 3 is a solution.
(d) The system has an infinite number of solutions.
(e) x 2, y 5, z 1 is not a solution.
In Questions 25 and 26, find the constants A, B, C that make
the statement true.
25.
x
2
4
x
x
7
6
x
A
3
x
B
2
REVIEW QUESTIONS
Year 1950 1960 1970 1980 1990 2000 2005
Miami 249.3 291.7 335.0 347.0 358.5 362.5 386.4
Cleveland 914.8 876.1 751.0 574.0 505.6 478.4 452.2
*U.S. Environmental Protection Agency.
†
U.S. Census Bureau.
CHAPTER 11 Review 821
26.
(x
6
2
x
2
1)
6
(
x
x
6
2)
x
A
1
x
B
1
x
C
2
In Questions 27–30, perform the indicated matrix multiplica-
tion or state that the product is not defined. Use these matrices.
10 2332
A
, B
, C
,
0 14124
12 23
E
34
, F
63
05 61
27. AB 28. CD 29. AE 30. DF
In Questions 31–34, find the inverse of the matrix, if it exists.
52 26
31.
32.
7 313
326 111
33.
112
34.
2 32
225 461
In Questions 35 and 36, use matrix inverses to solve the
system.
35. x 2z 6w 2
3x 4y 2z w 0
5x 2z 5w 4
4x 4y 2z 3w 1
36. 2x y 2z u 2
x 3y 4z 2u 2v 2
2x 3y 5z 4u v 1
x 2z 4v 4
2x 6z 5v 0
In Questions 37 and 38, find the equation of the parabola
passing through the given points.
37. (3, 52), (2, 17), (8, 305)
38. (2, 18), (2, 6), (4, 12)
39. The table shows the number of hours spent per person per
year on home video games.*
312
D
,
104
(a) Find a quadratic equation that models this data, with
x 0 corresponding to 1990.
(b) Use the equation to estimate the number of hours spent
in 2000 and 2015.
40. The table shows the per capita consumption of eggs in
selected years.*
(a) Find a quadratic equation that models this data, with
x 0 corresponding to 1980.
(b) Use the equation to estimate egg consumption in 1985,
1995, and 2005.
41. Tickets to a lecture cost $1 for students, $1.50 for faculty,
and $2 for others. Total attendance at the lecture was 460,
and the total income from tickets was $570. Three times as
many students as faculty attended. How many faculty mem-
bers attended the lecture?
42. If Andrew, Laura, and Ryan work together, they can paint a
large room in 4 hours. When only Laura and Ryan work
together, it takes 8 hours to paint the room. Andrew and
Laura, working together, take 6 hours to paint the room. How
long would it take each of them to paint the room alone?
[Hint: If x is the amount of the room painted in 1 hour by
Andrew, y is the amount painted by Laura, and z is the
amount painted by Ryan, then x y z 1/4.]
43. An animal feed is to be made from corn, soybeans, and meat
by-products. One bag is to supply 1800 units of fiber, 2800
units of fat, and 2200 units of protein. Each pound of corn
has 10 units of fiber, 30 units of fat, and 20 units of protein.
Each pound of soybeans has 20 units of fiber, 20 units of fat,
and 40 units of protein. Each pound of by-products has 30
units of fiber, 40 units of fat, and 25 units of protein. How
many pounds of corn, soybeans, and by-products should
each bag contain?
44. A company produces three camera models: A, B, and C.
Each model A requires 3 hours of lens polishing, 2 hours of
assembly time, and 2 hours of finishing time. Each model B
requires 2 hours of lens polishing, 2 hours of assembly time,
and 1 hour of finishing time. Each model C requires 1, 3,
and 1 hours of lens polishing, assembly, and finishing time,
respectively. There are 100 hours available for lens polish-
ing, 100 hours for assembly, and 65 hours for finishing each
week. How many of each model should be produced if all
available time is to be used?
Year 1999 2002 2004
Hours 58 70 77
Year 1980 1990 1999
Egg Consumption 271 234 255
*Data from Statistical Abstract of the United States: 2007.
*U.S. Department of Agriculture.
Chapter
11
Test
1. I paid $35.25 for 15 boxes of cereal. I bought some Basho
Bites, which cost $2.25 per box, and some Health Nuggets,
which cost $2.50 per box. How many boxes of each kind of
cereal did I buy?
2. Find the constants A, B, and C so that the following is an
identity:
x
A
4
7
x
B
2
x
3
C
3. Find a, b, c so the function f(x) ax
2
bx c conforms
to the following table of values.
4. Solve the following system of equations geometrically.
x y 8
3x 5y 0
5. Solve the following systems of equations:
(a) x 2y 1
3x 4y 4
(b) x y 5.0
3x 3y 7.8
05
0
3
x
y
5x
2
10 8x
x
3
4x
2
3x 12
(c) x
2
3 ln y 2
y e
x
6. Determine the number of solutions of the following
systems:
(a) x y 2
x y 8
(b) 5x 3y 61
15x 9y 18
(c) 2x y 3
x
1
2
y 4
(d) y sin x
y
4
x
0
2
7. Find two real numbers whose sum is 100 and whose
product is 2343.75.
8. Solve the following systems:
(a) 6x y 4z 41
y z 0
6z 12
(b) x y z 8
10x 3y 5z 4
2x 4y 2z 7
9. For the following matrices A, B, compute AB if it is defined,
and then compute BA if it is defined.
121300
(a) A
02 1
B
121
12 1 1 11
120
121
(b) A
130
B
211
423
(c) A
B
5
3
2
1
5
2
3
1
822 CHAPTER 11 Systems of Equations
xf(x)
03
14
23
823
Steve Cole/Getty Images
DISCOVERY PROJECT 11 Input-Output Analysis
Wassily Leontief won the Nobel Prize in economics in 1973 for his method of
input-output analysis of the economies of industrialized nations. This method has
become a permanent part of production planning and forecasting by both national
governments and private corporations. During the Arab oil boycott in 1973, for
example, General Electric used input-output analysis on 184 sectors of the econ-
omy (such as energy, agriculture, and transportation) to predict the effect of the
energy crisis on public demand for its products. The key to Leontief’s method is
knowing how much each sector of the economy needs from other sectors in order
to do its job.
A simple economic model will illustrate the basic ideas behind input-output
analysis. Suppose the country Hypothetica has just two sectors in its economy:
agriculture and manufacturing. The production of a ton of agricultural products
requires the use of .1 ton of agricultural products and .1 ton of manufactured prod-
ucts. Similarly, the production of a ton of manufactured products consumes .1 ton
of agricultural products and .3 ton of manufactured products. The key economic
question is: How many tons of each sector must Hypothetica produce in order to
have enough surplus to export 10,000 tons of agricultural products and 10,000
tons of manufactured goods?
824
DISCOVERY PROJECT 11
Let A be the total amount of agricultural goods, and let M be the total amount
of manufactured goods produced. The total agricultural production A is the sum of
the amount needed for producing the agricultural products plus the amount
needed for producing manufactured products plus the amount targeted for export.
In other words,
agricultural manufacturing export total agricultural
needs needs needs production
.1A .1M 100,000 A.
1. Write a similar equation for manufactured goods.
2. Solve the system of equations given by the agricultural equation above
and the manufacturing equation of Problem 1. What level of production
of agricultural and manufactured goods should Hypothetica work toward
to reach its export goals?
3. The same kind of analysis can be used to determine the surplus available
for exports in an economy. Suppose that Hypothetica can produce a total
of 120,000 tons of agricultural products and 28,000 tons of manufactured
goods. How much of each is available for export?
4. Input-output analysis can also be used for allocating resources in
smaller-scale situations. Suppose a horse outfitter is hired by the State of
Washington Department of Fish and Wildlife to haul 20 salt blocks into a
game area for winter feeding. Each horse can carry a 200-pound load. A
single person can handle three horses. Each person requires one horse to
ride and one-half horse to carry his or her personal gear. A single horse
can carry the feed and equipment for five horses. How many people and
horses must go on the trip? [Note: Each horse load of salt must consist of
an even number of blocks. Each salt block weighs 50 pounds. Also, frac-
tional numbers of horses and people are not allowed. Make sure to adjust
your answer to integer values, and be sure that the answer still provides
sufficient carrying capacity for the salt.]
DISCRETE ALGEBRA
What’s next?
CD and CD-ROM players, fax machines, cameras,
and other devices incorporate digital technology,
which uses sequences of 0’s and 1’s to send
signals. Determining the monthly payment on a car
loan involves the sum of a geometric sequence.
Problems involving the action of bouncing balls,
vacuum pumps, and other devices can sometimes
be solved by using sequences. See Exercises 55,
58, and 62 on page 851.
825
Chapter
© Terry Oakley/Alamy