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C. Determinants, Geometry, and the Coordinate Plane

# C. Determinants, Geometry, and the Coordinate Plane

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College Algebra G&M—

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689

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.

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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Section 7.4 Applications of Matrices and Determinants: Cramer’s Rule, Partial Fractions, and More

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

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

ruptured and is plummeting

back to Earth. Using an

onboard altimeter, the

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.

71. (4.3) Graph the polynomial using information

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

74. (4.3) Which is the graph (left or right) of a degree

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

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