10.E: Linear Equations (Optional Exercises)
- Page ID
- 10993
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
What are the dependent and independent variables?
Answer
dependent variable: fee amount; independent variable: time
Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
Graph the equation from Exercise.
Answer
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.
Find the equation that expresses the total fee in terms of the number of days the payment is late.
Graph the equation from Exercise.
Answer
Is the equation \(y = 10 + 5x – 3x^{2}\) linear? Why or why not?
Which of the following equations are linear?
- \(y = 6x + 8\)
- \(y + 7 = 3x\)
- \(y – x = 8x^{2}\)
- \(4y = 8\)
Answer
\(y = 6x + 8\), \(4y = 8\), and \(y + 7 = 3x\) are all linear equations.
Does the graph show a linear equation? Why or why not?
Table contains real data for the first two decades of AIDS reporting.
| 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 |
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 \(y = 50 + 100x\).
What are the independent and dependent variables?
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 \(y = 12,000x\).
What are the independent and dependent variables?
How many pounds of soil does the shoreline lose in a year?
Answer
12,000 pounds of soil
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.
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.
If you owned this stock, would you want a positive or negative slope? Why?