212 CHAPTER 3 Matrices
We solve this system using matrices, as follows: 1 1 1 500,000 1 1 1 500,000 ⫺0.08R ⫹ R S R 1 2 2 0.08 0.10 0.14 49,000 0 0.02 0.06 9,000 1 1 1 500,000 1 0 ⫺2 50,000 50R S R ⫺R ⫹ R S R 2 2 2 1 1 0 1 3 450,000 0 1 3 450,000
This matrix represents the equivalent system x ⫺ 2z ⫽ 50,000 so x ⫽ 50,000 ⫹ 2z
y ⫹ 3z ⫽ 450,000 y ⫽ 450,000 ⫺ 3z z ⫽ any real number
Because a negative amount cannot be invested, x, y, and z must al be nonnegative. Note
that x ⱖ 0 when z ⱖ 0 and that y ⱖ 0 when 450,000 ⫺ 3z ⱖ 0, or when z ⱕ 150,000. Thus the amounts invested are
$z at 14%, where 0 ⱕ z ⱕ 150,000
$x at 8%, where x ⫽ 50,000 ⫹ 2z
$y at 10%, where y ⫽ 450,000 ⫺ 3z
There are many possible investment plans. One example would be with z ⫽ $100,000,
x ⫽ $250,000, and y ⫽ $150,000. Observe that the values for x and y depend on z, and
once z is chosen, x wil be between $50,000 and $350,000, and y will be between $0 and$450,000. ■ ✓ CHECKPOINT 3 2 ⫺1 3 3 2 ⫺1 A N SW E R S 1. (a) 1 ⫺1 2 4 (b) 1 ⫺1 2 2 3 ⫺1 3 2 3 ⫺1 2. x ⫽ 1, y ⫽ 1, z ⫽ 2
3. x ⫽ 100, y ⫽ ⫺80, z ⫽ 110
4. x ⫽ z ⫹ 8, y ⫽ 5z ⫺ 2, z ⫽ any real number 5. No solution | EXERCISES | 3.3
In Problems 1 and 2, use the indicated row operation to
In Problems 5 and 6, each matrix is an augmented change matrix A, where
matrix used in the solution of a system of linear equa- 1
tions in x, y, and z. What is the solution of the system? ⫺2 ⫺1 ⫺7 A ⫽ 3 1 2 0 1 0 0 2 1 0 0 ⫺8 4 2 2 1 5. 0 1 0 12 6. 0 1 0 11 1. Add 0 0 1 ⫺5 0 0 1
⫺3 times row 1 to row 2 of matrix A and place 3
the result in row 2 to get 0 in row 2, column 1. 2. Add
In Problems 7–10, each matrix is an augmented matrix
⫺4 times row 1 to row 3 of matrix A and place
the result in row 3 to get 0 in row 3, column 1.
representing a system of linear equations in x, y, and z.
Use the Gauss-Jordan elimination method (see Gauss-
In Problems 3 and 4, write the augmented matrix
Jordan elimination method box and Example 1) to find
associated with each system of linear equations.
the solution of the system. x 1 1 2 ⫺1 1 5 2 6 ⫺ 3y ⫹ 4z ⫽ 2 x ⫹ 2y ⫹ 2z ⫽ 3 3. 2x 7. 0 3 1 7 8. 0 ⫺2 3 9 ⫹ 2z ⫽ 1 4. x ⫺ 2y ⫽ 4 x 0 ⫺2 4 0 0 1 3 0 ⫹ 2y ⫹ z ⫽ 1 y ⫺ z ⫽ 1
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SECTION 3.3 Gauss-Jordan Elimination: Solving Systems of Equations 213 1 2 5 ⫺4 1 1 1 3
The systems of equations in Problems 23–38 may have 9. 2 ⫺2 4 ⫺2 10. 3 ⫺2 4 5
unique solutions, an infinite number of solutions, or 0 1
no solution. Use matrices to find the general solution of ⫺3 7 1 2 1 4
each system, if a solution exists.
In Problems 11–16, use row operations on augmented x ⫹ y ⫹ z ⫽ 0 2x ⫺ y ⫹ 3z ⫽ 0
matrices to solve the given systems of linear equations. 23. 2x ⫺ y ⫺ z ⫽ 0 24. x ⫹ 2y ⫹ 2z ⫽ 0 x ⫹ y ⫺ z ⫽ 0 x ⫹ 2y ⫺ z ⫽ 3 ⫺x ⫹ 2y ⫹ 2z ⫽ 0 x ⫺ 3y ⫹ z ⫽ 0 11.
x ⫹ 2y ⫹ 3z ⫽ ⫺5 12. 2x ⫹ 5y ⫺ 2z ⫽ 7 3x ⫹ 2y ⫹ z ⫽ 0 x ⫹ 3y ⫹ 2z ⫽ 2 2x ⫺ y ⫺ 13z ⫽ 17 ⫺x ⫹ y ⫹ 5z ⫽ ⫺12 25.
x ⫹ y ⫹ 2z ⫽ 2 26. 2x ⫺ y ⫺ 2z ⫽ 1 2x ⫺ 6y ⫺ 12z ⫽ 6 2x ⫹ y ⫺ z ⫽ ⫺1 3x ⫹ 2y ⫽ 1 13. 3x ⫺ 10y ⫺ 20z ⫽ 5 2x ⫹ 2y ⫹ z ⫽ 2 2x ⫹ y ⫺ z ⫽ 2 2x ⫺ 17z ⫽ ⫺4 27.
x ⫺ 2y ⫹ 2z ⫽ 1 28. x ⫺ y ⫹ 2z ⫽ 3 ⫺3x ⫹ 6y ⫺ 9z ⫽ 3 ⫺x ⫹ 2y ⫺ 2z ⫽ ⫺1 x ⫹ y ⫺ z ⫽ 1 14. x ⫺ y ⫺ 2z ⫽ 0 x ⫺ 3y ⫹ 3z ⫽ 7 3x ⫹ 2y ⫺ 4z ⫽ ⫺12 5x ⫹ 5y ⫺ 7z ⫽ 63 29.
x ⫹ 2y ⫺ z ⫽ ⫺2 30. 3x ⫺ 3y ⫹ 2z ⫽ ⫺15 x ⫺ 2y ⫹ 3z ⫹ w ⫽ ⫺2 3x ⫹ 2y ⫹ 4z ⫽ 5 4x ⫹ 6y ⫹ z ⫽ 0 x 15. ⫺ 3y ⫹ z ⫺ w ⫽ ⫺7 2x ⫺ 5y ⫹ z ⫽ ⫺9 2x ⫺ y ⫹ z ⫽ 2 x ⫺ y ⫽ ⫺2 31.
x ⫹ 4y ⫺ 6z ⫽ 2 32. 3x ⫹ y ⫺ 6z ⫽ ⫺7 x ⫹ z ⫹ w ⫽ 2 3x ⫺ 4y ⫺ 2z ⫽ ⫺10 x ⫺ y ⫹ 2z ⫽ 3 x ⫹ 4y ⫺ 6z ⫺ 3w ⫽ 3 x y z 3 x y 33. ⫹ ⫹ ⫽ 34. 3 ⫹ 2 ⫹ z ⫽ 3 16. x ⫺ y ⫹ 2w ⫽ ⫺5 x ⫺ y ⫹ z ⫽ 4 x ⫺ y ⫺ z ⫽ 2 x ⫹ z ⫹ w ⫽ 1 3x ⫺ 2y ⫹ 5z ⫽ 14 x ⫺ 4y ⫹ z ⫽ ⫺4 y ⫹ z ⫹ w ⫽ 0 35. 36. 2x ⫺ 3y ⫹ 4z ⫽ 18 2x ⫺ 5y ⫺ 5z ⫽ ⫺9
In each of Problems 17–20, a system of linear equations
⫺0.6x ⫹ 0.1x ⫹ 0.4x ⫽ 10 1 2 3
and a reduced matrix for the system are given. (a) Use
37. 0.4x ⫺ 0.7x ⫹ 0.2x ⫽ ⫺26
the reduced matrix to find the general solution of the 1 2 3 0.2x ⫹ 0.6x ⫺ 0.5x ⫽ 20
system, if one exists. (b) If multiple solutions exist, find 1 2 3 two specific solutions. 0.1x ⫺ 0.1x ⫺ 0.3x ⫽ 17 1 2 3 x 38.
0.2x ⫹ 0.3x ⫹ 0.1x ⫽ ⫺9 ⫹ 2y ⫹ 3z ⫽ 1 1 0 3 1 2 3 5 0 17. 2x 0.3x ⫹ 0.2x ⫺ 0.5x ⫽ 5 ⫺ y ⫽ 3 0 1 6 1 2 3 5 0 x ⫹ 2y ⫹ 3z ⫽ 2 0 0 0 1
In Problems 39–46, use technology to solve each system 2x ⫹ 3y ⫹ 4z ⫽ 2 1 0 2 0
of equations, if a solution exists. 18. x ⫹ 2y ⫹ 2z ⫽ 1 0 1 0 0 x ⫹ 3x ⫹ 2x ⫹ 2x ⫽ 3 1 2 3 4 x ⫹ y ⫹ 2z ⫽ 2 0 0 0 1 39. x ⫹ x ⫹ 3x ⫽ 4 1 2 3 2x ⫹ y ⫺ z ⫽ 7 1 0 ⫺2 11 2x ⫹ 2x ⫺ 3x ⫽ 4 3 3 1 3 4 19. x ⫺ y ⫺ z ⫽ 4 0 1 1 x ⫺ 3x ⫽ 1 3 ⫺13 1 2 3x ⫹ 3y ⫺ z ⫽ 10 0 0 0 0 x ⫹ x ⫹ x ⫹ 2x ⫽ 1 1 2 3 4 3x ⫺ y ⫹ z ⫽ 3 1 0 2 7 x ⫺ 3x ⫽ 2 3 3 40. 1 3 20. 3x ⫹ 2z ⫽ 7 0 1 ⫺1 x ⫺ 3x ⫹ x ⫽ ⫺2 3 ⫺23 1 2 4 3x ⫺ 4y ⫹ 2z ⫽ 5 0 0 0 0 x ⫺ 4x ⫹ x ⫽ 0 2 3 4
21. How can you tel that a system of linear equations has 3x ⫹ 2y ⫹ z ⫺ w ⫽ 3
no solution by just looking at its reduced matrix? x ⫺ y ⫺ 2z ⫹ 2w ⫽ 2
22. Describe the procedure for finding the general solution 41. 2x ⫹ 3y ⫺ z ⫹ w ⫽ 1
of a system of linear equations if its reduced matrix
⫺x ⫹ y ⫹ 2z ⫺ 2w ⫽ ⫺2
indicates that it has an infinite number of solutions.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 214 CHAPTER 3 Matrices
2x ⫹ 3x ⫺ x ⫹ 3x ⫹ x ⫽ 22
52. Loans A bank lent $1.2 mil ion for the development 1 2 3 4 5 3x
of three new products, with one loan each at 6%, 7%,
⫹ x ⫺ 4x ⫹ 3x ⫺ x ⫽ 0 1 2 3 4 5 42. x
and 8%. The amount lent at 8% was equal to the sum of
⫹ x ⫹ 3x ⫺ 4x ⫹ 2x ⫽ 6 1 2 3 4 5
the amounts lent at the other two rates, and the bank’s
x ⫹ 2x ⫺ 3x ⫹ 2x ⫺ 2x ⫽ ⫺6 1 2 3 4 5
annual income from the loans was $88,000. How much 2x ⫹ 4x ⫺ 5x ⫽ ⫺7 1 4 5 was lent at each rate? x
53. Car rental patterns A car rental agency in a major city ⫹ 2x ⫺ x ⫹ x ⫽ 3 1 2 3 4
has a total of 2200 cars that it rents from three loca- 43. x ⫹ 3x ⫹ 4x ⫹ x ⫽ ⫺2 1 2 3 4
tions: Metropolis Airport, downtown, and the smal er 2x ⫹ 5x ⫹ 2x ⫹ 2x ⫽ 1 1 2 3 4
City Airport. Some weekly rental and return pat- 2x ⫹ 3x ⫺ 6x ⫹ 2x ⫽ 3 1 2 3 4
terns are shown in the table (note that Airport means x Metropolis Airport). ⫹ 2x ⫹ 3x ⫹ 4x ⫽ 2 1 2 3 4 44. x ⫹ 2x ⫹ 4x ⫹ 3x ⫽ 2 1 2 3 4 2x Rented from ⫹ 4x ⫹ 6x ⫹ 8x ⫽ 4 1 2 3 4 x Returned to AP DT CA ⫹ 4x ⫹ 3x ⫽ 4 1 2 3 Airport (AP) 90% 10% 10%
x ⫹ 3x ⫹ 4x ⫺ x ⫹ 2x ⫽ 1 1 2 3 4 5 Downtown (DT) 5% 80% 5% 45. x ⫺ x ⫺ 2x ⫹ x ⫽ 3 1 2 3 4
x ⫹ 2x ⫹ 3x ⫹ x ⫹ 4x ⫽ 0
At the beginning of a week, how many cars should be at 1 2 3 4 5 2x ⫹ 2x ⫹ 2x ⫹ 2x ⫽ 4
each location so that same number of cars wil be there 1 2 3 5
at the end of the week (and hence at the start of the
2x ⫺ x ⫹ x ⫹ 3x ⫹ 3x ⫽ 7 1 2 3 4 5 next week)? 46.
x ⫹ x ⫹ x ⫹ 2x ⫹ 2x ⫽ 0 1 2 3 4 5
54. Nutrition A psychologist studying the effects of
2x ⫹ x ⫺ x ⫹ x ⫹ x ⫽ ⫺3
nutrition on the behavior of laboratory rats is feeding 1 2 3 4 5
4x ⫺ x ⫺ 3x ⫹ x ⫹ x ⫽ 1
one group a combination of three foods: I, II, and III. 1 2 3 4 5
Each of these foods contains three additives, A, B, and a x y 47. Solve ⫹ b ⫽ c 1 1
1 for x. This gives a formula for
C, that are being used in the study. Each additive is a a x ⫹ b y ⫽ c 2 2 2
certain percentage of each of the foods as fol ows:
solving two equations in two variables for x. a x y Foods 48. Solve ⫹ b ⫽ c 1 1
1 for y. This gives a formula for a x ⫹ b y ⫽ c I II III 2 2 2
solving two equations in two variables for y. Additive A 10% 30% 60% Additive B 0% 4% 5% APPLICATIONS Additive C 2% 2% 12%
49. Nutrition A preschool has Campbel ’s Chunky Beef
soup, which contains 2.5 g of fat and 15 mg of choles-
If the diet requires 53 g per day of A, 4.5 g per day of B,
terol per serving (cup), and Campbel ’s Chunky Sirloin
and 8.6 g per day of C, find the number of grams per
Burger soup, which contains 7 g of fat and 15 mg of
day of each food that must be used.
cholesterol per serving. By combining the soups, it is
55. Nutrition The fol owing table gives the calories, fat,
possible to get 10 servings of soup that wil have 61 g of
and carbohydrates per ounce for three brands of cereal.
fat and 150 mg of cholesterol. How many cups of each
How many ounces of each brand should be com- soup should be used?
bined to get 443 calories, 5.7 g of fat, and 113.4 g of carbohydrates?
50. Ticket sales A 3500-seat theater sel s tickets for
$75and $110. Each night the theater’s expenses total
$245,000. When al 3500 seats sel , the owners want Cereal Calories Fat (g) Carbohydrates (g)
ticket revenues to cover expenses plus earn a profit All Fiber 50 0 22.0
of 25% of expenses. How many tickets of each price Frosted Puffs 108 0.1 25.7
should be sold to achieve this? Natural
51. Investment A man has $235,000 invested in three Mixed Grain 127 5.5 18.0
properties. One earns 12%, one 10%, and one 8%. His
annual income from the properties is $22,500 and the
56. Investment A brokerage house offers three stock port-
amount invested at 8% is twice that invested at 12%.
folios. Portfolio I consists of 2 blocks of common stock
(a) How much is invested in each property?
and 1 municipal bond. Portfolio II consists of 4 blocks
(b) What is the annual income from each property?
of common stock, 2 municipal bonds, and 3 blocks
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
SECTION 3.3 Gauss-Jordan Elimination: Solving Systems of Equations 215
of preferred stock. Portfolio III consists of 7 blocks of
The botanist wants to use a food that has these nutri-
common stock, 3 municipal bonds, and 3 blocks of pre-
ents in a different proportion and determines that he
ferred stock. A customer wants 21 blocks of common
wil need a total of 10,000 g of A, 20,000 g of B, and
stock, 10 municipal bonds, and 9 blocks of preferred
20,000 g of C. Find the number of bags of each type of
stock. How many units of each portfolio should be food that should be ordered. offered?
62. Traffic flow In the analysis of traffic flow, a certain city
57. Investment Suppose that portfolios I and II in Problem
estimates the fol owing situation for the “square” of its
56 are unchanged and portfolio III consists of 2 blocks
downtown district. In the fol owing figure, the arrows
of common stock, 2 municipal bonds, and 3 blocks of
indicate the flow of traffic. If x represents the num- 1
preferred stock. A customer wants 12 blocks of com-
ber of cars traveling between intersections A and B, x 2
mon stock, 6 municipal bonds, and 6 blocks of pre-
represents the number of cars traveling between B and
ferred stock. How many units of each portfolio should
C, x the number between C and D, and x the number 3 4 be offered?
between D and A, we can formulate equations based on
58. Nutrition Each ounce of substance A supplies 5% of
the principle that the number of vehicles entering an
the required nutrition a patient needs. Substance B
intersection equals the number leaving it. That is, for
supplies 15% of the required nutrition per ounce, and intersection A we obtain
substance C supplies 12% of the required nutrition
per ounce. If digestive restrictions require that sub- 200 ⫹ x ⫽ 100 ⫹ x 4 1
stances A and C be given in equal amounts and that
(a) Formulate equations for the traffic at B, C, and D.
the amount of substance B be one-fifth of these other
(b) Solve the system of these four equations.
amounts, find the number of ounces of each substance
(c) Can the entire traffic flow problem be studied
that should be in the meal to provide 100% nutrition.
and solved by counting only the x cars that travel
59. Nutrition A glass of skim milk supplies 0.1 mg of iron, 4 between A and D? Explain.
8.5 g of protein, and 1 g of carbohydrates. A quarter
pound of lean red meat provides 3.4 mg of iron, 22 g of In Out
protein, and 20 g of carbohydrates. Two slices of whole- 200 800
grain bread supply 2.2 mg of iron, 10 g of protein, and
12 g of carbohydrates. If a person on a special diet must
have 12.1 mg of iron, 97 g of protein, and 70 g of car- A x D Out 100 4 In 500
bohydrates, how many glasses of skim milk, how many
quarter-pound servings of meat, and how many two- x1 x3
slice servings of whole-grain bread wil supply this? In 300 Out 100
60. Transportation The King Trucking Company has an B x C 2
order for three products for delivery. The fol owing
table gives the particulars for the products. Out In Type I Type II Type III 200 200 Unit volume (cubic feet) 10 8 20 Unit weight (pounds) 10 20 40
63. Nutrition Three different bacteria are cultured in Unit value (dollars) 100 20 200
one dish and feed on three nutrients. Each individual
of species I consumes 1 unit of each of the first and
If the carrier can carry 6000 cu ft, can carry 11,000
second nutrients and 2 units of the third nutrient.
lb, and is insured for $36,900, how many units of each
Each individual of species II consumes 2 units of the type can be carried?
first nutrient and 2 units of the third nutrient. Each
61. Nutrition A botanist can purchase plant food of four
individual of species III consumes 2 units of the first
different types, I, II, III, and IV. Each food comes in the
nutrient, 3 units of the second nutrient, and 5 units of
same size bag, and the fol owing table summarizes the
the third nutrient. If the culture is given 5100 units of
number of grams of each of three nutrients that each
the first nutrient, 6900 units of the second nutrient, and bag contains.
12,000 units of the third nutrient, how many of each
species can be supported such that al of the nutrients Foods (grams) are consumed? I II III IV
64. Irrigation An irrigation system al ows water to flow
in the pattern shown in the figure below. Water flows Nutrient A 5 5 10 5
into the system at A and exits at B, C, D, and E with the Nutrient B 10 5 30 10
amounts shown. Using the fact that at each point the Nutrient C 5 15 10 25
water entering equals the water leaving, formulate an
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