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  • Chapter 2Linear Equations: Homework

    GRAPHING A LINEAR EQUATION

    Work the following problems.

    Is the point (2, 3) on the line 5x−2y=4?

    Yes
    Exercise 2.

    Is the point (1, – 2) on the line 6x−y=4?

    For the line 3x−y=12, complete the following ordered pairs.

    (2, ) ( , 6)

    (0, ) ( , 0)

    (2, –6) (6, 6)

    (0,–12) (4, 0)

    Exercise 4.

    For the line 4x+3y=24, complete the following ordered pairs.

    (3, ) ( , 4)

    (0, ) ( , 0)

    Graph y=2x+3

    graphics1.png
    Figure 2.0
    Exercise 6.

    Graph y=−3x+5

    Graph y=4x−3

    graphics2.png
    Figure 2.0
    Exercise 8.

    Graph x−2y=8

    Graph 2x+y=4

    graphics3.png
    Figure 2.0
    Exercise 10.

    Graph 2x−3y=6

    Graph 2x+4=0

    graphics4.png
    Figure 2.0
    Exercise 12.

    Graph 2y−6=0

    Graph the following three equations on the same set of coordinate axes.

    (2.1)y=x+1   y=2x+1  y=−x+1
    graphics5.png
    Figure 2.0
    Exercise 14.

    Graph the following three equations on the same set of coordinate axes.

    (2.2)y=2x+1  y=2x  y=2x−1

    Graph the line using the parametric equations

    (2.3)
    _autogen-svg2png-0015.png
    graphics6.png
    Figure 2.0
    Exercise 16.

    Graph the line using the parametric equations

    (2.4)x=2−3t,   y=1+2t

    SLOPE OF A LINE

    Find the slope of the line passing through the following pair of points.

    (2, 3) and (5, 9)

    (2.5)m=2
    Exercise 18.

    (4, 1) and (2, 5)

    (– 1, 1) and (1, 3)

    (2.6)m=1
    Exercise 20.

    (4, 3) and (– 1, 3)

    (6, – 5) and (4, – 1)

    (2.7)m=−2
    Exercise 22.

    (5, 3) and (– 1, – 4)

    (3, 4) and (3, 7)

    (2.8)m=undefined
    Exercise 24.

    (– 2, 4) and (– 3, – 2)

    (– 3, – 5) and (– 1, – 7)

    (2.9)m=−1
    Exercise 26.

    (0, 4) and (3, 0)

    Determine the slope of the line from the given equation of the line.

    (2.10)y=−2x+1
    (2.11)m=−2
    Exercise 28.

    y=3x−2

    (2.12)2x−y=6
    (2.13)m=2
    Exercise 30.
    (2.14)x+3y=6
    (2.15)3x−4y=12
    (2.16)m=3/4
    Exercise 32.

    What is the slope of the x-axis? How about the y-axis?

    Graph the line that passes through the given point and has the given slope.

    (1, 2) and m=−3/4

    graphics7.png
    Figure 2.0
    Exercise 34.

    (2, –1) and m=2/3

    (0, 2) and m=−2

    graphics8.png
    Figure 2.0
    Exercise 36.

    (2, 3) and m=0

    DETERMINING THE EQUATION OF A LINE

    Write an equation of the line satisfying the following conditions. Write the equation in the form _autogen-svg2png-0034.png.

    It passes through the point (3, 10) and has slope=2.

    (2.17)y=2x+4
    Exercise 38.

    It passes through the point (4,5) and has m=0.

    It passes through (3, 5) and (2, – 1).

    (2.18)y=6x−13
    Exercise 40.

    It has slope 3, and its y-intercept equals 2.

    It passes through (5, – 2) and m=2/5.

    (2.19)y=2/5x−4
    Exercise 42.

    It passes through (– 5, – 3) and (10, 0).

    It passes through (4, – 4) and (5, 3).

    (2.20)y=7x−32
    Exercise 44.

    It passes through (7, – 2) and its y-intercept is 5.

    It passes through (2, – 5) and its x-intercept is 4.

    (2.21)y=5/2x−10
    Exercise 46.

    Its a horizontal line through the point (2, – 1).

    It passes through (5, – 4) and (1, – 4).

    (2.22)y=−4
    Exercise 48.

    It is a vertical line through the point (3, – 2).

    It passes through (3, – 4) and (3, 4).

    (2.23)x=3
    Exercise 50.

    It has x-intercept=3 and y-intercept=4.

    Write an equation of the line satisfying the following conditions. Write the equation in the form _autogen-svg2png-0047.png.

    It passes through (3, – 1) and m=2.

    (2.24)2x−y=7
    Exercise 52.

    It passes through (– 2, 1) and m=−3/2.

    It passes through (– 4, – 2) and m=3/4.

    (2.25)3x−4y=−4
    Exercise 54.

    Its x-intercept equals 3, and m=−5/3.

    It passes through (2, – 3) and (5, 1).

    (2.26)4x−3y=17
    Exercise 56.

    It passes through (1, – 3) and (– 5, 5).

    APPLICATIONS

    In the following application problems, assume a linear relationship holds.

    The variable cost to manufacture a product is $25, and the fixed costs are $1200. If x represents the number of items manufactured and y the cost, write the cost function.

    (2.27)y=25x+1200
    Exercise 58.

    It costs $90 to rent a car driven 100 miles and $140 for one driven 200 miles. If x is the number of miles driven and y the total cost of the rental, write the cost function.

    The variable cost to manufacture an item is $20, and it costs a total of $750 to produce 20 items. If x represents the number of items manufactured and y the cost, write the cost function.

    (2.28)y=20x+350
    Exercise 60.

    To manufacture 30 items, it costs $2700, and to manufacture 50 items, it costs $3200. If x represents the number of items manufactured and y the cost, write the cost function.

    To manufacture 100 items, it costs $32,000, and to manufacture 200 items, it costs $40,000. If x represents the number of items manufactured and y the cost, write the cost function.

    (2.29)y=80x+24000
    Exercise 62.

    It costs $1900 to manufacture 60 items, and the fixed costs are $700. If x represents the number of items manufactured and y the cost, write the cost function.

    A person who weighs 150 pounds has 60 pounds of muscles, and a person that weighs 180 pounds has 72 pounds of muscles. If x represents the body weight and y the muscle weight, write an equation describing their relationship. Use this relationship to determine the muscle weight of a person that weighs 170 pounds.

    _autogen-svg2png-0072.png
    Exercise 64.

    A spring on a door stretches 6 inches if a force of 30 pounds is applied, and it stretches 10 inches if a force of 50 pounds is applied. If x represents the number of inches stretched, and y the force applied, write an equation describing the relationship. Use this relationship to determine the amount of force required to stretch the spring 12 inches.

    A male college student who is 64 inches tall weighs 110 pounds, and another student who is 74 inches tall weighs 180 pounds. Assuming the relationship between male students' heights (x), and weights (y) is linear, write a function to express weights in terms of heights, and use this function to predict the weight of a student who is 68 inches tall.

    y=7x−338 ; 138
    Exercise 66.

    EZ Clean company has determined that if it spends $30,000 on advertisement, it can hope to sell 12,000 of its Minivacs a year, but if it spends $50,000, it can sell 16,000. Write an equation that gives a relationship between the number of dollars spent on advertisement (x) and the number of minivacs sold (y).

    The freezing temperatures for Celsius and Fahrenheit scales are 0 degree and 32 degrees, respectively. The boiling temperatures for Celsius and Fahrenheit are 100 degrees and 212 degrees, respectively. Let C denote the temperature in Celsius and F in Fahrenheit. Write the conversion function from Celsius to Fahrenheit, and use this function to convert 25 degrees Celsius into an equivalent Fahrenheit measure.

    F=9/5C+32; 77ºF
    Exercise 68.

    By reversing the coordinates in the previous problem, find a conversion function that converts Fahrenheit into Celsius, and use this conversion function to convert 72 degrees Fahrenheit into an equivalent Celsius measure.

    The population of California in the year 1960 was 17 million, and in 1995 it was 32 million. Write the population function, and use this function to find the population of California in the year 2010. (Hint: Use the year 1960 as the base year, that is, assume 1960 as the year zero. This will make 1995, and 2010 as the years 35, and 50, respectively.)

    _autogen-svg2png-0083.png
    Exercise 70.

    In the U. S. the number of people infected with the HIV virus in 1985 was 1,000, and in 1995 that number became 350,000. If the increase in the number is linear, write an equation that will give the number of people infected in any year. If this trend continues, what will the number be in 2010? (Hint: See previous problem.)

    In 1975, an average house in San Jose cost $45,000 and the same house in 1995 costs $195,000. Write an equation that will give the price of a house in any year, and use this equation to predict the price of a similar house in the year 2010.

    y=7500x+45000; $307,500
    Exercise 72.

    An average math text book cost $25 in 1980, and $60 in 1995. Write an equation that will give the price of a math book in any given year, and use this equation to predict the price of the book in 2010.

    MORE APPLICATIONS

    Solve the following problems.

    Solve for x and y.
    y=3x+4
    y=5x−2

    x=3, y=13
    Exercise 74.

    Solve for x and y.
    2x−3y=4
    3x−4y=5

    The supply curve for a product is y=2000x+13000, and the demand curve is y=−1000x+28000, where x represents the price and y the number of items. At what price will the supply equal demand, and how many items will be produced at that price?

    x=5, y=23000
    Exercise 76.

    The supply curve for a product is y=300x+9000, and the demand curve is y=−100x+14000, where x represents the price and y the number of items. At what price will the supply equal demand, and how many items will be produced at that price?

    A demand curve for a commodity is the number of items the consumer will buy at different prices. It has been determined that at a price of $2 a store can sell 2400 of a particular type of toy dolls, and for a price of $8 the store can sell 600 such dolls. If x represents the price of dolls and y the number of items sold, write an equation for the demand curve.

    (2.30)y=−300x+3000
    Exercise 78.

    A supply curve for a commodity is the number of items of the product that can be made available at different prices. A manufacturer of toy dolls can supply 2000 dolls if the dolls are sold for $8 each, but he can supply only 800 dolls if the dolls are sold for $2 each. If x represents the price of dolls and y the number of items, write an equation for the supply curve.

    The equilibrium price is the price where the supply equals the demand. From the demand and supply curves obtained in the previous two problems, find the equilibrium price, and determine the number of items that can be sold at that price.

    (2.31)x=5.2 y=1440
    Exercise 80.

    A car rental company offers two plans. Plan I charges $10 a day and 10 cents a mile, while Plan II charges 14 cents a mile, but no flat fee. If you were to drive 300 miles in a day, which plan is better? For what mileage are both rates equal?

    A break-even point is the intersection of the cost function and the revenue function, that is, where the total cost equals revenue. Mrs. Jones Cookies Store's revenue and cost in dollars for x number of cookies is given by R=.80x and C=.05x+3000. Find the number of cookies that must be sold so that the revenue and cost are the same.

    (2.32)x=4000
    Exercise 82.

    A company's revenue and cost in dollars are given by R=225x and C=75x+6000, where x is the number of items. Find the number of items that must be produced to break-even.

    A firm producing computer diskettes has a fixed costs of $10,725, and variable cost of 20 cents a diskette. Find the break-even point if the diskettes sell for $1.50 each.

    (8250, 12375)
    Exercise 84.

    Whackemhard Sports is planning to introduce a new line of tennis rackets. The fixed costs for the new line are $25,000 and the variable cost of producing each racket is $60. If the racket sells for $80, find the number of rackets that must be sold in order to break even.

    CHAPTER REVIEW

    Find an equation of the x-axis.

    (2.33)y=0

    Find the slope of the line whose equation is 2x+3y=6.

    (2.34)−2/3

    Find the slope of the line whose equation is y=−3x+5.

    –3

    Find both the x and y intercepts of the line 3x−2y=12.

    4, –6

    Find an equation of the line whose slope is 3 and y-intercept 5.

    (2.35)y=3x+5

    Find an equation of the line whose x-intercept is 2 and y-intercept 3.

    (2.36)3x+2y=6

    Find an equation of the line that has slope 3 and passes through the point (2, 15).

    (2.37)y=3x+9

    Find an equation of the line that has slope _autogen-svg2png-0125.png and passes through the point (4, 3).

    (2.38)3x+2y=18

    Find an equation of the line that passes through the points (0, 32) and (100, 212).

    (2.39)y=9/5x+32

    Find an equation of the line that passes through the point (2, 5) and is parallel to the line y=3x+4.

    (2.40)y=3x−1

    Find the point of intersection of the lines 2x−3y=9 and 3x+4y=5.

    (3, –1)

    Is the point (3, – 2) on the line 5x−2y=11?

    No

    Find two points on the line given by the parametric equations, x=2+3t, y=1−2t.

    (2,1), (5,–1); Answers will vary

    Find two points on the line 2x−6=0.

    (3,0), (3, 1); Answers will vary

    Graph the line 2x−3y+6=0.

    The line through (–3,0) and (0,2)

    Graph the line y=−2x+3.

    The line through (0,3) and (1,1)

    A female college student who is 60 inches tall weighs 100 pounds, and another female student who is 66 inches tall weighs 124 pounds. Assuming the relationship between the female students' weights and heights is linear, write an equation giving the relationship between heights and weights of female students, and use this relationship to predict the weight of a female student who is 70 inches tall.

    (2.41)
    _autogen-svg2png-0138.png

    In deep-sea diving, the pressure exerted by water plays a great role in designing underwater equipment. If at a depth of 10 feet there is a pressure of 21 lb/in2, and at a depth of 50 ft there is a pressure of 75 lb/in2, write an equation giving a relationship between depth and pressure. Use this relationship to predict pressure at a depth of 100 ft.

    (2.42)
    _autogen-svg2png-0139.png

    If the variable cost to manufacture an item is $30, and the fixed costs are $2750, write the cost function.

    (2.43)y=30x+2750

    The variable cost to manufacture an item is $10, and it costs $2,500 to produce 100 items. Write the cost function, and use this function to estimate the cost of manufacturing 300 items.

    (2.44)
    _autogen-svg2png-0141.png

    It costs $2,700 to manufacture 100 items of a product, and $4,200 to manufacture 200 items. If x represents the number of items, and y the costs, find the cost equation, and use this function to predict the cost of 1,000 items.

    (2.45)
    _autogen-svg2png-0142.png

    In 1980, the average house in Palo Alto cost $280,000 and the same house in 1997 costs $450,000. Assuming a linear relationship, write an equation that will give the price of the house in any year, and use this equation to predict the price of a similar house in the year 2010.

    (2.46)
    _autogen-svg2png-0143.png

    The population of Argentina in 1987 was 31.5 million and in 1997 it was 42.5 million. Assuming a linear relationship, write an equation that will give the population of Argentina in any year, and use this equation to predict the population of Argentina in the year 2010.

    (2.47)
    _autogen-svg2png-0144.png

    In 1955, an average new Chevrolet sold for $2,400, and a similar Chevrolet sold for $15,000 in 1995. Assuming a linear relationship, write an equation that will give the price of an average Chevrolet in any year. Use this equation to predict the price of an average Chevrolet in the year 2010.

    (2.48)
    _autogen-svg2png-0145.png

    Two-hundred items are demanded at a price of $5, and 300 items are demanded at a price of $3. If x represents the price, and y the number of items, write the demand function.

    (2.49)y=−50x+450

    A supply curve for a product is the number of items of the product that can be made available at different prices. A manufacturer of Tickle Me Elmo dolls can supply 2000 dolls if the dolls are sold for $30 each, but he can supply only 800 dolls if the dolls are sold for $10 each. If x represents the price of dolls and y the number of items, write an equation for the supply curve.

    (2.50)y=60x+200

    Suppose you are trying to decide on a price for your latest creation - a coffee mug that never tips. Through a survey, you have determined that at a price of $2, you can sell 2100 mugs, but at a price of $12 you can only sell 100 mugs. Furthermore, your supplier can supply you 3500 mugs if you charge your customers $12, but only 500 if you charge $2. What price should you charge so that the supply equals demand, and at that price how many coffee mugs will you be able to sell?

    (2.51)Price=$5.20; # of mugs=1460

    A car rental company offers two plans. Plan I charges $12 a day and 12 cents a mile, while Plan II charges $30 a day but no charge for miles. If you were to drive 300 miles in a day, which plan is better? For what mileage are both rates the same?

    150 miles

    The supply curve for a product is y=250x−1,000 and the demand curve for the same product is y=−350x+8,000, where x is the price and y the number of items produced. Find the following.

    1. At a price of $10, how many items will be in demand?

    2. At what price will 4,000 items be supplied?

    3. What is the equilibrium price for this product?

    4. How many items will be manufactured at the equilibrium price?

    1. 4500;

    2. 20;

    3. 15;

    4. 2750

    The supply curve for a product is y=625x−600 and the demand curve for the same product is y=−125x+8,400, where x is the price and y the number of items produced. Find the equilibrium price and determine the number of items that will be produced at that price.

    $12; 6900

    Both Jenny and Masur are sales people for Athletic Shoes. Jenny gets paid $8 per hour plus 4% commission on the sales. Masur gets paid $10 per hour plus 8% commission on the sales in excess of $1,000. If they work 8-hour days, for what sales amount would they both earn the same daily amounts?

    1600

    A company's revenue and cost in dollars are given by R=25x and C=10x+9,000, where x represents the number of items. Find the number of items that must be produced to break-even.

    600

    A firm producing video tapes has fixed costs of $6,800, and a variable cost of 30 cents per tape. If the video tapes sell for $2 each, find the number of tapes that must be produced to break-even.

    4,000

    A firm producing disposable cameras has fixed costs of $8,000, and variable cost of 50 cents a camera. If the cameras sell for $3.50, how many cameras must be produced to break-even?

    2,667

    The Stanley Company is coming up with a new cordless travel shaver just before the Christmas holidays . It hopes to sell 10,000 of these shavers in the month of December alone. The manufacturing variable cost is $3 and the fixed costs $100,000. If the shavers sell for $11 each, how many must be produced to break-even?

    12,500
    Solutions