C. Determinants, Geometry, and the Coordinate Plane
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Section 7.4 Applications of Matrices and Determinants: Cramer’s Rule, Partial Fractions, and More
7.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. The determinant `
as:
2.
a11
a21
a12
` is evaluated
a22
rule uses a ratio of determinants to
solve for the unknowns in a system.
x1
y1
1
x3
y3
1
collinear if 0T 0 ϭ † x2 y2 1 † has a value of
.
3. Given the matrix of coefficients D, the matrix Dx is
formed by replacing the coefficients of x with the
terms.
ᮣ
4. The three points (x1, y1), (x2, y2), and (x3, y3) are
5. Discuss/Explain the process of writing
.
8x Ϫ 3
as a
x2 Ϫ x
sum of partial fractions.
6. Discuss/Explain why Cramer’s rule cannot be
applied if D ϭ 0. Use an example to illustrate.
DEVELOPING YOUR SKILLS
Write the determinants D, Dx, and Dy for the systems
given, but do not solve.
7. e
2x ϩ 5y ϭ 7
Ϫ3x ϩ 4y ϭ 1
8. e
Ϫx ϩ 5y ϭ 12
3x Ϫ 2y ϭ Ϫ8
Solve each system of equations using Cramer’s rule, if
possible. Do not use a calculator.
9. e
4x ϩ y ϭ Ϫ11
3x Ϫ 5y ϭ Ϫ60
10. e
x ϭ Ϫ2y Ϫ 11
y ϭ 2x Ϫ 13
y
x
ϩ ϭ1
8
4
11. μ
y
x
ϭ ϩ6
5
2
3
7
2
xϪ yϭ
3
8
5
12. μ
5
3
11
xϩ yϭ
6
4
10
13. e
0.6x Ϫ 0.3y ϭ 8
0.8x Ϫ 0.4y ϭ Ϫ3
14. e
Ϫ2.5x ϩ 6y ϭ Ϫ1.5
0.5x Ϫ 1.2y ϭ 3.6
The two systems given in Exercises 15 and 16 are identical except for the third equation. For the first system given,
(a) write the determinants D, Dx, Dy, and Dz then (b) determine if a solution using Cramer’s rule is possible by
computing ͦ Dͦ without the use of a calculator (do not solve the system). Then (c) compute ͦDͦ for the second system
and try to determine how the equations in the second system are related.
4x Ϫ y ϩ 2z ϭ Ϫ5
4x Ϫ y ϩ 2z ϭ Ϫ5
15. • Ϫ3x ϩ 2y Ϫ z ϭ 8 , • Ϫ3x ϩ 2y Ϫ z ϭ 8
x Ϫ 5y ϩ 3z ϭ Ϫ3
xϩ yϩ zϭ3
2x ϩ
3z ϭ Ϫ2
2x ϩ
3z ϭ Ϫ2
16. • Ϫx ϩ 5y ϩ z ϭ 12 , • Ϫx ϩ 5y ϩ z ϭ 12
3x Ϫ 2y ϩ z ϭ Ϫ8
x ϩ 5y ϩ 4z ϭ 10
Use Cramer’s rule to solve each system of equations. Verify computations using a graphing calculator.
x ϩ 2y ϩ 5z ϭ 10
17. • 3x ϩ 4y Ϫ z ϭ 10
x Ϫ y Ϫ z ϭ Ϫ2
x ϩ 3y ϩ 5z ϭ 6
18. • 2x Ϫ 4y ϩ 6z ϭ 14
9x Ϫ 6y ϩ 3z ϭ 3
y ϩ 2z ϭ 1
19. • 4x Ϫ 5y ϩ 8z ϭ Ϫ8
8x Ϫ 9z ϭ 9
x ϩ 2y ϩ 5z ϭ 10
20. • 3x Ϫ z ϭ 8
Ϫy Ϫ z ϭ Ϫ3
w ϩ 2x Ϫ 3y ϭ Ϫ8
x Ϫ 3y ϩ 5z ϭ Ϫ22
21. μ
4w Ϫ 5x ϭ 5
Ϫy ϩ 3z ϭ Ϫ11
w Ϫ 2x ϩ 3y Ϫ z ϭ 11
3w Ϫ 2y ϩ 6z ϭ Ϫ13
22. μ
2x ϩ 4y Ϫ 5z ϭ 16
3x Ϫ 4z ϭ 5
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CHAPTER 7 Matrices and Matrix Applications
DECOMPOSITION OF RATIONAL EXPRESSIONS
Exercises 23 through 32 are designed solely to reinforce the various possibilities for decomposing a rational expression.
All are proper fractions whose denominators are completely factored. Set up the decomposition template using
appropriate numerators, but do not solve.
23.
3x ϩ 2
1x ϩ 321x Ϫ 22
24.
27.
x2 ϩ 5
x1x Ϫ 321x ϩ 12
28.
31.
x3 ϩ 3x Ϫ 2
1x ϩ 121x2 ϩ 22 2
32.
Ϫ4x ϩ 1
1x Ϫ 221x Ϫ 52
x2 Ϫ 7
1x ϩ 421x Ϫ 22x
25.
29.
2x3 ϩ 3x2 Ϫ 4x ϩ 1
x1x2 ϩ 32 2
3x2 Ϫ 2x ϩ 5
1x Ϫ 121x ϩ 221x Ϫ 32
26.
Ϫ2x2 ϩ 3x Ϫ 4
1x ϩ 321x ϩ 121x Ϫ 22
x2 ϩ x Ϫ 1
x2 1x ϩ 22
30.
x2 Ϫ 3x ϩ 5
1x Ϫ 321x ϩ 22 2
Decompose each rational expression into partial fractions.
33.
4Ϫx
x2 ϩ x
34.
3x ϩ 13
x ϩ 5x ϩ 6
35.
2x Ϫ 27
2x ϩ x Ϫ 15
36.
Ϫ11x ϩ 6
5x Ϫ 4x Ϫ 12
37.
8x2 Ϫ 3x Ϫ 7
x3 Ϫ x
38.
x2 ϩ 24x Ϫ 12
x3 Ϫ 4x
39.
3x2 ϩ 7x Ϫ 1
x3 ϩ 2x2 ϩ x
40.
Ϫ2x2 Ϫ 7x ϩ 28
x3 Ϫ 4x2 ϩ 4x
41.
3x2 ϩ 10x ϩ 4
8 Ϫ x3
42.
3x2 ϩ 4x Ϫ 1
x3 Ϫ 1
43.
6x2 ϩ x ϩ 13
x3 ϩ 2x2 ϩ 3x ϩ 6
44.
2x2 Ϫ 14x Ϫ 7
x3 Ϫ 2x2 ϩ 5x Ϫ 10
45.
x4 Ϫ x2 Ϫ 2x ϩ 1
x5 ϩ 2x3 ϩ x
46.
Ϫ3x4 ϩ 13x2 ϩ x Ϫ 12
x5 ϩ 4x3 ϩ 4x
47.
x3 Ϫ 17x2 ϩ 76x Ϫ 98
1x2 Ϫ 6x ϩ 92 1x2 Ϫ 2x Ϫ 32
48.
16x3 Ϫ 66x2 ϩ 98x Ϫ 54
12x2 Ϫ 3x2 14x2 Ϫ 12x ϩ 92
ᮣ
2
2
2
WORKING WITH FORMULAS
L r2
† . The determinant shown can be used to find the area of a Norman
؊
W
2
W
window (rectangle ؉ half-circle) with length L, width W, and radius r ؍. Find the area of the following windows.
2
Area of a Norman window: A † ؍
49.
50.
16 in.
32 cm
20 in.
ᮣ
58 cm
APPLICATIONS
Geometric Applications
Find the area of the triangle with the vertices given. Assume units are in centimeters.
51. (2, 1), (3, 7), and (5, 3)
52. 1Ϫ2, 32, 1Ϫ3, Ϫ42, and 1Ϫ6, 12
Find the area of the parallelogram with vertices given. Assume units are in feet.
53. 1Ϫ4, 22, 1Ϫ6, Ϫ12, 13, Ϫ12, and 15, 22
54. 1Ϫ5, Ϫ62 , (5, 0), (5, 4), and 1Ϫ5, Ϫ22
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691
The volume of a triangular pyramid is given by the formula V ؍13Bh, where B represents the area of the triangular
base and h is the height of the pyramid. Find the volume of a triangular pyramid whose height is given and whose
base has the coordinates shown. Assume units are in meters.
55. h ϭ 6 m; vertices (3, 5), 1Ϫ4, 22, and 1Ϫ1, 62
Determine if the following sets of points are collinear.
57. (1, 5), 1Ϫ2, Ϫ12 , and (4, 11)
59. 1Ϫ2.5, 5.22, 11.2, Ϫ5.62 , and 12.2, Ϫ8.52
56. h ϭ 7.5 m; vertices 1Ϫ2, 32, 1Ϫ3, Ϫ42, and 1Ϫ6, 12
58. (1, 1), 13, Ϫ52, and 1Ϫ2, 92
60. 1Ϫ0.5, 2.552, 1Ϫ2.8, 1.632, and (3, 3.95)
For each linear equation given, substitute the first two points to verify they are solutions. Then use the test for
collinear points to determine if the third point is also a solution.
61. 2x Ϫ 3y ϭ 7; 12, Ϫ12, 1Ϫ1.3, Ϫ3.22, 1Ϫ3.1, Ϫ4.42
62. 5x ϩ 2y ϭ 4; 12, Ϫ32, 13.5, Ϫ6.752, 1Ϫ2.7, 8.752
Write a linear system that models each application. Then solve using Cramer’s rule.
63. Return on investments: If $15,000 is invested at a
certain interest rate and $25,000 is invested at
another interest rate, the total return was $2900. If
the investments were reversed the return would be
$2700. What was the interest rate paid on each
investment?
64. Cost of fruit: Many years ago, two pounds of
apples, 2 lb of kiwi, and 10 lb of pears cost $3.26.
Three pounds of apples, 2 lb of kiwi, and 7 lb of
pears cost $2.98. Two pounds of apples, 3 lb of
kiwi, and 6 lb of pears cost $2.89. Find the cost of
a pound of each fruit.
65. Forces on trusses of a
roof: Triangular trusses
have been used for
decades in the
construction of homes,
bridges, tower supports,
and other projects. If we
consider a very simple truss
in the form of an equilateral
60°
triangle, the forces
exerted along the
rafters of the truss by F1
F2
a weight at the apex
can be modeled by
a 2 ϫ 2 system of
linear equations.
60°
60°
If a 180-lb
carpenter is working at the center of this truss, the
forces along each rafter can be modeled by the
system shown. Find the force along each rafter.
23
1F1 ϩ F2 2 ϭ 180
• 2
F1 Ϫ F2 ϭ 0
66. Dietary research for
pets: As part of a
research project, a
college student is
mixing a special diet
for pet mice from two available sources. The
diet must offer exactly 22.8 g of protein and 5 g
of fat. Given the protein and fat values for the
food sources shown, how much of each should
be used?
Source 1
Source 2
protein value
0.18
0.24
fat value
0.06
0.04
67. High-altitude weather
research: A high-altitude
weather balloon carrying a
heavy payload has suddenly
ruptured and is plummeting
back to Earth. Using an
onboard altimeter, the
payload radios its height in
feet every 2 sec after
rupture. For data of the
form (time in seconds, height in feet), three of the
readings are (5, 9600), (10, 8400), and (15, 6400).
(a) Use these data to find an equation of the form
h ϭ at2 ϩ bt ϩ c that models the height of the
balloon at any time t. (b) At what height did
the balloon rupture? (c) What is the altitude of
the balloon after 20 sec? (d) How many seconds
until the payload hits the ground?
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68. Manufacturing
surfboards:
Australian
Waterglide is a
manufacturer of
custom surfboards
for beginners,
recreational
surfers, and surfers participating in international
competitions. For each board, production is handled
in three stages: forming, fiberglass, and finishing.
ᮣ
The number of hours required for each stage are
given in the table. If the company has 80 labor hours
per week available for forming, 152 hr available for
fiberglass, and 145 hr available for finishing, how
many boards of each type should be made?
Beginner
forming
Recreational
Competition
3
4
5
fiberglass
4.5
8.5
9
finishing
5.5
7.5
8
EXTENDING THE CONCEPT
69. Find the area of the pentagon whose vertices are:
1Ϫ5, Ϫ52, 15, Ϫ52, 18, 62, 1Ϫ8, 62, and (0, 12.5).
ᮣ
7–56
CHAPTER 7 Matrices and Matrix Applications
70. The polynomial form for the equation of a circle is
x2 ϩ y2 ϩ Dx ϩ Ey ϩ F ϭ 0. Find the equation of
the circle that contains the points 1Ϫ1, 72, (2, 8),
and 15, Ϫ12.
MAINTAINING YOUR SKILLS
71. (4.3) Graph the polynomial using information
about end-behavior, y-intercept, x-intercept(s), and
midinterval points: f 1x2 ϭ x3 Ϫ 2x2 Ϫ 7x ϩ 6.
73. (5.3/5.5) Solve the equation 32xϪ1 ϭ 92Ϫx two
ways. First using logarithms, then by equating the
bases and using properties of equality.
72. (2.2) Which is the graph (left or right) of
g1x2 ϭ ϪͿx ϩ 1Ϳ ϩ 3? Justify your answer.
74. (4.3) Which is the graph (left or right) of a degree
3 polynomial? Justify your answer.
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
7.5
Ϫ5Ϫ4 Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
10
8
6
4
2
10
8
6
4
2
5
4
3
2
1
1 2 3 4 5 x
y
y
y
y
5
4
3
2
1
1 2 3 4 5 x
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
1 2 3 4 5 x
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
1 2 3 4 5 x
Matrix Applications and Technology Use
LEARNING OBJECTIVES
In Section 7.5 you will see
how we can:
Most of the skills needed for this study have been presented in previous sections. Here
we’ll use various types of regression, combined with systems of equations, to solve
practical applications from business and industry.
A. Use matrix equations to
solve static systems
B. Use matrices for
encryption/decryption
A. Solving Static Systems with Varying Constraints
When the considerations of a business or industry involve more than two variables,
solutions using matrix methods have a distinct advantage over other methods. Companies often have to perform calculations using basic systems weekly, daily, or even
hourly, to keep up with trends, market changes, changes in cost of raw materials, and
so on. In many situations, the basic requirements remain the same, but the frequently
changing inputs require a recalculation each time they change.
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Section 7.5 Matrix Applications and Technology Use
EXAMPLE 1
ᮣ
693
Determining Supply Inventories Using Matrices
BNN Soft Drinks receives new orders daily for its most popular drink, Saratoga
Cola. It can deliver the carbonated beverage in a twelve-pack of 12-ounce (oz)
cans, a six-pack of 20-oz bottles, or in a 2-L bottle. The ingredients required to
produce a twelve-pack include 1 gallon (gal) of carbonated water, 1.25 pounds (lb)
of sugar, 2 cups (c) of flavoring, and 0.5 grams (g) of caffeine. For the six-pack,
0.8 gal of carbonated water, 1 lb of sugar, 1.6 c of flavoring, and 0.4 g of caffeine
are needed. The 2-L bottle contains 0.47 gal of carbonated water, 0.59 lb of sugar,
0.94 c of flavoring, and 0.24 g of caffeine. How much of each ingredient must be
on hand for Monday’s order of 300 twelve-packs, 200 six-packs, and 500 2-L
bottles? What quantities must be on hand for Tuesday’s order: 410 twelve-packs,
320 six-packs, and 275 2-L bottles?
Solution
ᮣ
Begin by setting up a general system of equations, letting x represent the number
of twelve-packs, y the number of six-packs, and z the number of 2-L bottles:
1x ϩ 0.8y ϩ 0.47z ϭ gallons of carbonated water
1.25x ϩ 1y ϩ 0.59z ϭ pounds of sugar
μ 2x ϩ 1.6y ϩ 0.94z ϭ cups of flavoring
0.5x ϩ 0.4y ϩ 0.24z ϭ grams of caffeine
As a matrix equation we have
1 0.8
1.25 1
2 1.6
0.5 0.4
0.47
w
x
s
0.59
Ơ ÊyĐ ¥
0.94
f
z
c
0.24
Enter the 4 ϫ 3 matrix as matrix A, and the size of
the order as matrix as B. Using a calculator, we find
1
1.25
AB
2
0.5
0.8
1
1.6
0.4
695
0.47
300
870
0.59
Ơ,
Ơ Ê 200 Đ
1390
0.94
500
350
0.24
and BNN Soft Drinks will need 695 gal of
410
carbonated water, 870 lb of sugar, 1390 c of
flavoring, and 350 g of caffeine for Monday’s order. After entering C ϭ £ 320 §
275
for Tuesday’s orders, computing the product AC shows 795.25 gal of carbonated
water, 994.75 lb of sugar, 1590.5 c of flavoring, and 399 g of caffeine are needed
for Tuesday.
Now try Exercises 1 through 6
ᮣ
Example 1 showed how the creation of a static matrix can help track and control
inventory requirements. In Example 2, we use a static matrix to solve a system that will
identify the amount of data traffic used by a company during various hours of the day.
EXAMPLE 2
ᮣ
Identifying the Source of Data Traffic Using Matrices
Mariño Imports is a medium-size company that is considering upgrading from a
1.544 megabytes per sec (Mbps) T1 Internet line to a fractional T3 line with a
bandwidth of 7.72 Mbps. They currently use their bandwidth for phone traffic, office
data, and Internet commerce. The IT (Information Technology) director devises a plan
to monitor how much data traffic each resource uses on an hourly basis. Because of