SECTION 1-2 Systems of Linear Equations and Applications

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motivation (pull) and position was given approximately by the equation 1

p 5d 70

Systems of Linear Equations and Applications

17

If the same rat were trained as described in this problem and in Problem 73, at what distance (to one decimal place) from the goal box would the approach and avoidance strengths be the same? (What do you think the rat would do at this point?)

30 d 170

where pull p is measured in grams and distance d in centimeters. When the pull registered was 40 grams, how far was the rat from the goal box? Puzzle

74. Professor Brown performed the same kind of experiment as described in Problem 73, except that he replaced the food in the goal box with a mild electric shock. With the same kind of apparatus, he was able to measure the avoidance strength relative to the distance from the object to be avoided. He found that the avoidance strength a (measured in grams) was related to the distance d that the rat was from the shock (measured in centimeters) approximately by the equation

75. An oil-drilling rig in the Gulf of Mexico stands so that onefifth of it is in sand, 20 feet of it is in water, and two-thirds of it is in the air. What is the total height of the rig? 76. During a camping trip in the North Woods in Canada, a couple went one-third of the way by boat, 10 miles by foot, and one-sixth of the way by horse. How long was the trip? ★★

4

a 3d 230

SECTION

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30 d 170

77. After exactly 12 o’clock noon, what time will the hands of a clock be together again?

Systems of Linear Equations and Applications • Systems of Equations • Substitution • Applications

• Systems of

Equations

In the preceding section, we solved word problems by introducing a single variable representing one of the unknown quantities and then tried to represent all other unknown quantities in terms of this variable. In certain word problems, it is more convenient to introduce several variables, find equations relating these variables, and then solve the resulting system of equations. For example, if a 12-foot board is cut into two pieces so that one piece is 2 feet longer than the other, then letting x Length of the longer piece y Length of the shorter piece we see that x and y must satisfy both the following equations: x y 12 xy2 We now have a system of two linear equations in two variables. Thus, we can solve this problem by finding all pairs of numbers x and y that satisfy both equations. In general, we are interested in solving linear systems of the type: ax by h cx dy k

System of two linear equations in two variables

18

1 Equations and Inequalities

where x and y are variables and a, b, c, d, h, and k are real constants. A pair of numbers x x0 and y y0 is a solution of this system if each equation is satisfied by the pair. The set of all such pairs of numbers is called the solution set for the system. To solve a system is to find its solution set. In this section, we will restrict our discussion to simple solution techniques for systems of two linear equations in two variables. Larger systems and more sophisticated solution methods will be discussed in Chapter 8.

• Substitution

EXAMPLE 1

To solve a system by substitution, we first choose one of the two equations in a system and solve for one variable in terms of the other. (We make a choice that avoids fractions, if possible.) Then we substitute the result in the other equation and solve the resulting linear equation in one variable. Finally, we substitute this result back into the expression obtained in the first step to find the second variable. We illustrate this process by returning to the board problem stated at the beginning of the section.

Solving a System by Substitution Solve the board problem by solving the system x y 12 xy2

Solution

Solve either equation for one variable and substitute into the remaining equation. We choose to solve the first equation for y in terms of x: x y 12 y 12 x aEEbEEc

Solve the first equation for y in terms of x. Substitute into the second equation.

xy2 x (12 x) 2 x 12 x 2 2x 14 x7 Now, replace x with 7 in y 12 x: y 12 x y 12 7 y5 Thus, the longer board measures 7 feet and the shorter board measures 5 feet.

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Check

Matched Problem 1


x y 12

xy2

7 5 ‚ 12

75‚2

12 ⁄ 12

2⁄2

Solve by substitution and check: x y 3 x 2y 3

EXAMPLE 2

Solving a System by Substitution Solve by substitution and check: 2x 3y 7 3x y 7

Solution

To avoid fractions, we choose to solve the second equation for y: 3x y 7

Solve for y in terms of x.

y 3x 7 y 3x 7 2x 3y 7

Substitute into first equation. First equation

2x 3(3x 7) 7

Solve for x.

2x 9x 21 7 7x 14 x2

Substitute x 2 in y 3x 7.

y 3x 7 y 3(2) 7 y 1 Thus, the solution is x 2 and y 1. Check

Matched Problem 2

2x 3y 7

3x y 7

2(2) 3(1) ‚ 7

3(2) (1) ‚ 7

7⁄7

7⁄7

Solve by substitution and check: 3x 4y 18 2x y 1

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EXPLORE-DISCUSS 1

• Applications

EXAMPLE 3

Use substitution to solve each of the following systems. Discuss the nature of the solution sets you obtain. x 3y 4

x 3y 4

2x 6y 7

2x 6y 8

The following examples illustrate the advantages of using systems of equations and substitution in solving word problems.

Diet An individual wants to use milk and orange juice to increase the amount of calcium and vitamin A in her daily diet. An ounce of milk contains 38 milligrams of calcium and 56 micrograms* of vitamin A. An ounce of orange juice contains 5 milligrams of calcium and 60 micrograms of vitamin A. How many ounces of milk and orange juice should she drink each day to provide exactly 550 milligrams of calcium and 1,200 micrograms of vitamin A?

Solution

First we define the relevant variables: x Number of ounces of milk y Number of ounces of orange juice Next we summarize the given information in a table. It is convenient to organize the tables so that the quantities represented by the variables correspond to columns in the table (rather than to rows), as shown.

Milk

Orange Juice

Total Needed

Calcium

38

05

0,550

Vitamin A

56

60

1,200

Now we use the information in the table to form equations involving x and y: in x oz Calcium of milk 38x

in y oz Calcium of orange juice

05y

calcium Total needed (mg)

0,550

A in x oz A in y oz vitamin A Vitamin Vitamin Total of milk of orange juice needed (g) 56x

60y

*A microgram (g) is one-millionth (106) of a gram.

1,200

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5y 550 38x

21

Solve first equation for y.

y 110 7.6x

(1)

56x 60(110 7.6x) 1,200

Substitute for y in second equation.

56x 6,600 456x 1,200 400x 5,400 x 13.5

Substitute in (1).

y 110 7.6(13.5) y 7.4 Drinking 13.5 ounces of milk and 7.4 ounces of orange juice each day will provide the required amounts of calcium and vitamin A. Check

Matched Problem 3

EXAMPLE 4 San 2,400 Francisco miles Washington, D.C.

Solution

38x 5y 550

56x 60y 1,000

38(13.5) 5(7.4) ‚ 500

56(13.5) 60(7.4) ‚ 1,200

500 ⁄ 500

1,200 ⁄ 1,200

An individual wants to use cottage cheese and yogurt to increase the amount of protein and calcium in his daily diet. An ounce of cottage cheese contains 3 grams of protein and 12 milligrams of calcium. An ounce of yogurt contains 1 gram of protein and 44 milligrams of calcium. How many ounces of cottage cheese and yogurt should he eat each day to provide exactly 57 grams of protein and 840 milligrams of calcium?

Airspeed An airplane makes the 2,400-mile trip from Washington, D.C., to San Francisco in 7.5 hours and makes the return trip in 6 hours. Assuming that the plane travels at a constant airspeed and that the wind blows at a constant rate from west to east, find the plane’s airspeed and the wind rate.

Let x represent the airspeed of the plane and let y represent the rate at which the wind is blowing (both in miles per hour). The ground speed of the plane is determined by combining these two rates; that is, x y Ground speed flying east to west (headwind) x y Ground speed flying west to east (tailwind) Applying the familiar formula D RT to each leg of the trip leads to the following system of equations: 2,400 7.5(x y)

From Washington to San Francisco

2,400 6(x y)

From San Francisco to Washington

22


After simplification, we have x y 320 x y 400 Solve using substitution: x y 320

Solve first equation for x.

y 320 y 400

Substitute in second equation.

2y 80 y 40 mph

Wind rate

x 40 320 x 360 mph

Check

Matched Problem 4

EXAMPLE 5

Airspeed

2,400 7.5(x y)

2,400 6(x y)

2,400 ‚ 7.5(360 40)

2,400 ‚ 6(360 40)

2,400 ⁄ 2,400

2,400 ⁄ 2,400

A boat takes 8 hours to travel 80 miles upstream and 5 hours to return to its starting point. Find the speed of the boat in still water and the speed of the current.

Supply and Demand The quantity of a product that people are willing to buy during some period of time depends on its price. Generally, the higher the price, the less the demand; the lower the price, the greater the demand. Similarly, the quantity of a product that a supplier is willing to sell during some period of time also depends on the price. Generally, a supplier will be willing to supply more of a product at higher prices and less of a product at lower prices. The simplest supply and demand model is a linear model. Suppose we are interested in analyzing the sale of cherries each day in a particular city. Using special analytic techniques (regression analysis) and collected data, an analyst arrives at the following price–demand and price–supply equations: p 0.3q 5

Demand equation (consumer)

p 0.06q 0.68

Supply equation (supplier)

where q represents the quantity in thousands of pounds and p represents the price in dollars. For example, we see that consumers will purchase 11 thousand pounds (q 11) when the price is p 0.3(11) 5 $1.70 per pound. On the other hand, suppliers will be willing to supply 17 thousand pounds of cherries at $1.70 per pound (solve 1.7 0.06q 0.68 for q). Thus, at $1.70 per pound the suppliers are willing

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to supply more cherries than the consumers are willing to purchase. The supply exceeds the demand at that price, and the price will come down. At what price will cherries stabilize for the day? That is, at what price will supply equal demand? This price, if it exists, is called the equilibrium price, and the quantity sold at that price is called the equilibrium quantity. How do we find these quantities? We solve the linear system p 0.3q 5

Demand equation

p 0.06q 0.68

Supply equation

using substitution (substituting p 0.3q 5 into the second equation). 0.3q 5 0.06q 0.68 0.36q 4.32 q 12 thousand pounds

(equilibrium quantity)

Now substitute q 12 back into either of the original equations in the system and solve for p (we choose the second equation): p 0.06(12) 0.68 p $1.40 per pound

Matched Problem 5

(equilibrium price)

The price–demand and price–supply equations for strawberries in a certain city are p 0.2q 4

Demand equation

p 0.04q 1.84

Supply equation

where q represents the quantity in thousands of pounds and p represents the price in dollars. Find the equilibrium quantity and the equilibrium price.

EXAMPLE 6

Cost and Revenue A publisher is planning to produce a new textbook. The fixed costs (reviewing, editing, typesetting, and so on) are $320,000, and the variable costs (printing, sales commissions, and so on) are $31.25 per book. The wholesale price (the amount received by the publisher) will be $43.75 per book. How many books must the publisher sell to break even; that is, so that costs will equal revenues?

Solution

If x represents the number of books printed and sold, then the cost and revenue equations for the publisher are y 320,000 31.25x

Cost equation

y 43.75x

Revenue equation

24


The publisher breaks even when costs equal revenues. We can find when this occurs by solving this system. Using the second equation to substitute for y in the first equation, we have 43.75x 320,000 31.25x 12.5x 320,000 x 25,600 Thus, the publisher will break even when 25,600 books are printed and sold.

Matched Problem 6

A computer software company is planning to market a new word processor. The fixed costs (design, programming, and so on) are $720,000, and the variable costs (disk duplication, manual production, and so on) are $25.40 per copy. The wholesale price of the word processor will be $44.60 per copy. How many copies of the word processor must the company manufacture and sell to break even?

Answers to Matched Problems 1. x 1, y 2 2. x 2, y 3 3. 13.9 oz cottage cheese, 15.3 oz of yogurt 4. Boat: 13 mph; current: 3 mph 5. Equilibrium quantity 9 thousand pounds; Equilibrium price $2.20 per pound 6. 37,500 copies

EXERCISE

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A

11. y 5.46x

Solve Problems 1–6 by substitution. 1. y 3x 5

2. y 5x 8

y 4x 7

y 2x 4

4. 2x 7y 4 x 3y 1

5. 6x 11y 16 2x 3y 8

3. 4x 5y 8 3x y 5 6. 9x 7y 8 4x 3y 27

B Solve Problems 7–16 by substitution. 7. 3s 5t 30 7s 11t 32 9. 13m 3n 10 7m 3n 10

8. 2s 9t 22 8s 11t 48 10. 12m 11n 2 18m 7n 3

12. y 7.15x

y 6,300 4.86x

y 13,860 3.85x

13. 0.7u 0.2v 0.5

14. 0.4u 0.9v 0.71

0.3u 0.5v 0.09 15. 65a 32b 4 1 3a

54b 1

0.8u 0.2v 0.3 16. 38a 12b 2 9 10 a

25b 4

17. In the process of solving a system by substitution, suppose you encounter a contradiction, such as 0 1. How would you describe the solutions to such a system? Illustrate your ideas with the system x 2y 3 2x 4y 7 18. In the process of solving a system by substitution, suppose you encounter an identity, such as 0 0. How would you

1-2

describe the solutions to such a system? Illustrate your ideas with the system x 2y 3 2x 4y 6

C In Problems 19 and 20, solve each system for p and q in terms of x and y. Explain how you could check your solution and then perform the check. 19. x 2 p 2q

20. x 1 2p q

y 3 p 3q

y 4 p q

Problems 21 and 22 refer to the system ax by h cx dy k where x and y are variables and a, b, c, d, h, and k are real constants. 21. Solve the system for x and y in terms of the constants a, b, c, d, h, and k. Clearly state any assumptions you must make about the constants during the solution process. 22. Discuss the nature of solutions to systems that do not satisfy the assumptions you made in Problem 21.

APPLICATIONS

23. Airspeed. It takes a private airplane 8.75 hours to make the 2,100-mile flight from Atlanta to Los Angeles and 5 hours to make the return trip. Assuming that the wind blows at a constant rate from Los Angeles to Atlanta, find the airspeed of the plane and the wind rate. 24. Airspeed. A plane carries enough fuel for 20 hours of flight at an airspeed of 150 miles per hour. How far can it fly into a 30 mph headwind and still have enough fuel to return to its starting point? (This distance is called the point of no return.) –Time. A crew of eight can row 20 kilometers per 25. Rate– hour in still water. The crew rows upstream and then returns to its starting point in 15 minutes. If the river is flowing at 2 km/h, how far upstream did the crew row? –Time. It takes a boat 2 hours to travel 20 miles down 26. Rate– a river and 3 hours to return upstream to its starting point. What is the rate of the current in the river? 27. Chemistry. A chemist has two solutions of hydrochloric acid in stock: a 50% solution and an 80% solution. How much of each should be used to obtain 100 milliliters of a 68% solution?


25

28. Business. A jeweler has two bars of gold alloy in stock, one of 12 carats and the other of 18 carats (24-carat gold is pure 12 gold, 12-carat is 24 pure, 18-carat gold is 18 24 pure, and so on). How many grams of each alloy must be mixed to obtain 10 grams of 14-carat gold? 29. Break-Even Analysis. It costs a small recording company $17,680 to prepare a compact disc. This is a one-time fixed cost that covers recording, package design, and so on. Variable costs, including such things as manufacturing, marketing, and royalties, are $4.60 per CD. If the CD is sold to music shops for $8 each, how many must be sold for the company to break even? 30. Break-Even Analysis. A videocassette manufacturer has determined that its weekly cost equation is C 3,000 10x, where x is the number of cassettes produced and sold each week. If cassettes are sold for $15 each to distributors, how many must be sold each week for the manufacturer to break even? (Refer to Problem 29.) 31. Finance. Suppose you have $12,000 to invest. If part is invested at 10% and the rest at 15%, how much should be invested at each rate to yield 12% on the total amount invested? 32. Finance. An investor has $20,000 to invest. If part is invested at 8% and the rest at 12%, how much should be invested at each rate to yield 11% on the total amount invested? 33. Production. A supplier for the electronics industry manufactures keyboards and screens for graphing calculators at plants in Mexico and Taiwan. The hourly production rates at each plant are given in the table. How many hours should each plant be operated to exactly fill an order for 4,000 keyboards and screens?

Plant

Keyboards

Screens

Mexico

40

32

Taiwan

20

32

34. Production. A company produces Italian sausages and bratwursts at plants in Green Bay and Sheboygan. The hourly production rates at each plant are given in the table. How many hours should each plant be operated to exactly fill an order for 62,250 Italian sausages and 76,500 bratwursts?

Italian sausage

Bratwurst

Green Bay

800

0,800

Sheboygan

500

1,000

Plant

26


35. Nutrition. Animals in an experiment are to be kept on a strict diet. Each animal is to receive, among other things, 20 grams of protein and 6 grams of fat. The laboratory technician is able to purchase two food mixes of the following compositions: Mix A has 10% protein and 6% fat; mix B has 20% protein and 2% fat. How many grams of each mix should be used to obtain the right diet for a single animal? 36. Nutrition. A fruit grower can use two types of fertilizer in an orange grove, brand A and brand B. Each bag of brand A contains 8 pounds of nitrogen and 4 pounds of phosphoric acid. Each bag of brand B contains 7 pounds of nitrogen and 7 pounds of phosphoric acid. Tests indicate that the grove needs 720 pounds of nitrogen and 500 pounds of phosphoric acid. How many bags of each brand should be used to provide the required amounts of nitrogen and phosphoric acid? ★

37. Supply and Demand. At $0.60 per bushel, the daily supply for wheat is 450 bushels and the daily demand is 645 bushels. When the price is raised to $0.90 per bushel, the daily supply increases to 750 bushels and the daily demand decreases to 495 bushels. Assume that the supply and demand equations are linear. (A) Find the supply equation. [Hint: Write the supply equation in the form p aq b and solve for a and b.] (B) Find the demand equation. (C) Find the equilibrium price and quantity.

★

38. Supply and Demand. At $1.40 per bushel, the daily supply for soybeans is 1,075 bushels and the daily demand is 580 bushels. When the price falls to $1.20 per bushel, the daily supply decreases to 575 bushels and the daily demand increases to 980 bushels. Assume that the supply and demand equations are linear. (A) Find the supply equation. [See the hint in Problem 37.] (B) Find the demand equation. (C) Find the equilibrium price and quantity.

SECTION

1-3

★

39. Physics. An object dropped off the top of a tall building falls vertically with constant acceleration. If s is the distance of the object above the ground (in feet) t seconds after its release, then s and t are related by an equation of the form s a bt2 where a and b are constants. Suppose the object is 180 feet above the ground 1 second after its release and 132 feet above the ground 2 seconds after its release. (A) Find the constants a and b. (B) How high is the building? (C) How long does the object fall?

★

40. Physics. Repeat Problem 39 if the object is 240 feet above the ground after 1 second and 192 feet above the ground after 2 seconds.

★

41. Earth Science. An earthquake emits a primary wave and a secondary wave. Near the surface of the Earth the primary wave travels at about 5 miles per second and the secondary wave at about 3 miles per second. From the time lag between the two waves arriving at a given receiving station, it is possible to estimate the distance to the quake. (The epicenter can be located by obtaining distance bearings at three or more stations.) Suppose a station measured a time difference of 16 seconds between the arrival of the two waves. How long did each wave travel, and how far was the earthquake from the station?

★

42. Earth Science. A ship using sound-sensing devices above and below water recorded a surface explosion 6 seconds sooner by its underwater device than its above-water device. Sound travels in air at about 1,100 feet per second and in seawater at about 5,000 feet per second. (A) How long did it take each sound wave to reach the ship? (B) How far was the explosion from the ship?

Linear Inequalities • Inequality Relations and Interval Notation • Solving Linear Inequalities • Applications We now turn to the problem of solving linear inequalities in one variable, such as 3(x 5) 5(x 7) 10

and

4 3 2x 7