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12.2E: Linear Equations (Exercises)

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    4689
  • Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

    Exercise 12.2.5

    What are the dependent and independent variables?

    Answer

    dependent variable: fee amount; independent variable: time

    Exercise 12.2.6

    Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

    Exercise 12.2.7

    Graph the equation from Exercise.

    Answer

    This is a graph of the equation y = 25 + 12.50x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 25 from 0 - 100. The equation's graph is a line that crosses the y-axis at 25 and is sloped up and to the right, rising 12.50 units for every one unit of run.

    Figure 12.2.4.

    Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

    Exercise 12.2.8

    Find the equation that expresses the total fee in terms of the number of days the payment is late.

    Exercise 12.2.9

    Graph the equation from Exercise.

    Answer

    This is a graph of the equation y = 10 + 5x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 10 from 0 - 50. The equation's graph is a line that crosses the y-axis at 10 and is sloped up and to the right, rising 5 units for every one unit of run.

    Figure 12.2.5.

    Exercise 12.2.10

    Is the equation \(y = 10 + 5x – 3x^{2}\) linear? Why or why not?

    Exercise 12.2.11

    Which of the following equations are linear?

    1. \(y = 6x + 8\)
    2. \(y + 7 = 3x\)
    3. \(y – x = 8x^{2}\)
    4. \(4y = 8\)

    Answer

    \(y = 6x + 8\), \(4y = 8\), and \(y + 7 = 3x\) are all linear equations.

    Exercise 12.2.12

    Does the graph show a linear equation? Why or why not?

    This is a graph of an equation. The x-axis is labeled in intervals of 1 from -5 to 5; the y-axis is labeled in intervals of 1 from 0 - 8. The equation's graph is a parabola, a u-shaped curve that has a minimum value at (0, 0).

    Figure 12.2.6.

    Table contains real data for the first two decades of AIDS reporting.

    Adults and Adolescents only, United States
    Year # AIDS cases diagnosed # AIDS deaths
    Pre-1981 91 29
    1981 319 121
    1982 1,170 453
    1983 3,076 1,482
    1984 6,240 3,466
    1985 11,776 6,878
    1986 19,032 11,987
    1987 28,564 16,162
    1988 35,447 20,868
    1989 42,674 27,591
    1990 48,634 31,335
    1991 59,660 36,560
    1992 78,530 41,055
    1993 78,834 44,730
    1994 71,874 49,095
    1995 68,505 49,456
    1996 59,347 38,510
    1997 47,149 20,736
    1998 38,393 19,005
    1999 25,174 18,454
    2000 25,522 17,347
    2001 25,643 17,402
    2002 26,464 16,371
    Total 802,118 489,093

    Exercise 12.2.13

    Use the columns "year" and "# AIDS cases diagnosed. Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?

    Answer

    The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable.

    Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is \(y = 50 + 100x\).

    Exercise 12.2.14

    What are the independent and dependent variables?

    Exercise 12.2.15

    What is the y-intercept and what is the slope? Interpret them using complete sentences.

    Answer

    The \(y\)-intercept is 50 (\(a = 50\)). At the start of the cleaning, the company charges a one-time fee of $50 (this is when \(x = 0\)). The slope is 100 (\(b = 100\)). For each session, the company charges $100 for each hour they clean.

    Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y = 12,000x\).

    Exercise 12.2.16

    What are the independent and dependent variables?

    Exercise 12.2.17

    How many pounds of soil does the shoreline lose in a year?

    Answer

    12,000 pounds of soil

    Exercise 12.2.18

    What is the \(y\)-intercept? Interpret its meaning.

    Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is \(y = 15 – 1.5x\) where \(x\) is the number of hours passed in an eight-hour day of trading.

    Exercise 12.2.19

    What are the slope and y-intercept? Interpret their meaning.

    Answer

    The slope is -1.5 (\(b = -1.5\)). This means the stock is losing value at a rate of $1.50 per hour. The \(y\)-intercept is $15 (\(a = 15\)). This means the price of stock before the trading day was $15.

    Exercise 12.2.19

    If you owned this stock, would you want a positive or negative slope? Why?