(Source: From 2011 World Almanac. Reprinted by permission.) 32. Earnings and Race The table gives the median house-
hold income for whites and blacks in various years. a. Let x represent the median household income for
whites and y represent the corresponding median household income for blacks, and make a scatter plot of these data.
b. Find a linear model that expresses the median household income for blacks as a function of the median household income for whites.
c. Find the slope of the linear model in (b) and write a sentence that interprets it.
33. Poverty The table shows the number of millions of people in the United States who lived below the pov- erty level for selected years.
a. Find a linear model that approximately fits the data, using x as the number of years after 1970. b. Use a graph of the model and a scatter plot to
determine if the model is nearly an exact fit for the data.
Year Population (millions) Year Population (millions) 2000 275.3 2060 432.0 2010 299.9 2070 463.6 2020 324.9 2080 497.8 2030 351.1 2090 533.6 2040 377.4 2100 571.0 2050 403.7 (Source: www.census.gov/population/projections)
b. Does it appear that a line will be a reasonable fit for the data?
c. Find the linear model which is the best fit for the data.
d. Use the unrounded model to estimate the U.S. per- sonal consumption for 2012.
Year
Persons Living Below the Poverty Level (millions) 1970 25.4 1975 25.9 1980 29.3 1986 32.4 1990 33.6 1995 36.4 2000 31.1 2004 37.0 2005 37.0 2006 36.5 2007 37.2 2008 39.8
(Source: U.S. Census Bureau, U.S. Department of Commerce)
Year CPI 1970 38.8 1980 82.4 1990 130.7 2000 172.2 2005 195.3 2010 217.5
(Source: Bureau of Labor Statistics)
Year Personal Consumption ($ billions) 1990 3835.5 1995 4987.3 2000 6830.4 2005 8819.0 2007 9806.3 2008 10,104.5 2009 10,001.3
(Source: U.S. Department of Commerce) 34. Consumer Price Index Prices as measured by the
U.S. Consumer Price Index (CPI) have risen steadily since World War II. The data in the table give the CPI for selected years between 1970 and 2010. The CPI in this table has 1984 as a reference year; that is, what cost $1 in 1984 cost about $1.31 in 1990 and $2.18 in 2010.
a. Align the input data as the number of years after 1970 and find a linear model for the data rounded to three decimal places.
b. Use the model to estimate when the CPI will be 262.39.
35. Personal Consumption The sum of the personal con- sumption expenditures in the United States, in bil- lions of dollars, for selected years from 1990 through 2009 is shown in the table that follows.
a. Make a scatter plot of the data, with x equal to the number of years past 1990 and y equal to the bil- lions of dollars spent.
36. U.S. Population The following table gives projec- tions of the U.S. population from 2000 to 2100. a. Find a linear function that models the data, with x
equal to the number of years after 2000 and f (x) equal to the population in millions.
b. Find f (65) and state what it means.
c. What does this model predict the population to be in 2080? How does this compare with the value for 2080 in the table?
37. Gross Domestic Product The table on the next page gives the gross domestic product (the value of all goods and services, in billions of dollars) of the United States for selected years from 1970 to 2009. a. Create a scatter plot of the data, with y represent-
ing the GDP in billions of dollars and x represent- ing the number of years after 1970.
39. Online Marketing Online marketing is emerging as an increasingly effective way for Hollywood to reach its target audience. In the past five years, Hollywood b. Find the linear function that best fits the data, with
x equal to the number of years after 1970.
c. Graph the model with the scatter plot to see if the line is a good fit for the data.
38. Smoking The table gives the percent of adults aged 18 and over in the United States who reported smok- ing for selected years.
a. Write the equation that is the best fit for the data, with x equal to the number of years after 1990. b. What does the model estimate the percent to be in
2012?
c. When will the percent be 16, according to the model?
Year Number Imprisoned
1980 13,766 1985 20,605 1990 28,659 1995 31,805 2000 50,451 2005 61,151 2006 63,699 2007 62,883
(Source: Bureau of Justice)
Year Gross Domestic Product
1970 1038.3 1980 2788.1 1990 5800.5 2000 9951.5 2005 12,638.4 2007 14,061.8 2008 14,369.1 2009 14,119.0
(Source: U.S. Bureau of Economic Analysis)
Year Smoking 1999 23.5 2000 23.2 2001 22.7 2002 22.4 2003 21.6 2004 20.9 2005 20.9 2006 20.8 2007 19.7 2008 20.5 2009 20.6
(Source: Centers for Disease Control)
Year Online Ad Spending ($ millions) 2006 259 2007 370 2008 508 2009 650 2010 760 2011 857 (Source: eMarketer)
has placed more emphasis on new media channels, as shown in the table below.
a. Write the linear equation that models online ad spending as a function of the number of years after 2000.
b. According to the model, what is the annual increase in the amount of online spending?
c. According to the model, what is the percent increase in the amount of online spending from 2009 to 2010?
d. What is the average rate of change of spending from 2006 to 2011?
40. Prison Sentences The proportion of those convicted in federal court who are imprisoned has been increas- ing. The table shows the number of those imprisoned for selected years from 1980 to 2007.
a. Use the data to create a linear equation that mod- els the number of those imprisoned as a function of the number of years after 1980.
b. Use the model to estimate the number imprisoned in 2002. Is this interpolation or extrapolation? c. Use the model to determine in what year the num-
Worldwide Box Office Revenue (US$ billions)
2005 2006 2007 2008 2009
U.S./Canada 8.8 9.2 9.1 9.6 10.6
International 14.3 16.3 16.6 18.1 19.3
Worldwide 23.1 25.5 26.3 27.8 29.9
(Source: Motion Picture Association of America)
Weight (lb) Usual Dosage (mg)
88 40 99 45 110 50 121 55 132 60 143 65 154 70 165 75 176 80 187 85 198 90
Weight not over: Parcel Post Rate
1 lb $5.15 2 5.38 3 6.39 4 7.14 5 8.28 (Source: USPS) 1980 1989 1998 Year 2007 20,000 0 60,000 40,000 80,000 100,000 Total Convicted Imprisoned
Defendants in Cases Concluded in U.S. District Courts
41. Drug Doses The table below shows the usual dosage for a certain prescription drug that combats bacterial infections for a person’s weight.
b. Compare the outputs of the model with the data outputs from the table for several values in the table. Is the model a perfect fit?
a. Find a linear function D = f (W) that models the dosage given in the table as a function of the patient’s weight.
b. Compare the outputs of the model with the data out- puts from the table for several values in the table. How well does the model fit the data?
c. What does the model give as the dosage for a 150- pound person?
d. Should this model be interpreted discretely or continuously?
42. Parcel Post Postal Rates The table that follows gives U.S. postage rates for local parcel post mail for zone 3. Each given weight refers to the largest weight package that can be mailed for the corresponding postage. a. Find a linear function P = f (W ) that models the
postage in the table as a function of the weight in the table.
43. Box Office Revenue Worldwide box office revenue for all films reached $29.9 billion in 2009, up 7.6% over the 2008 total. U.S./Canada and international box office revenues in U.S. dollars were both up sig- nificantly over the 2005 total.
a. Let x = the number of years after 2000 and draw a scatter plot of the U.S./Canada data.
b. Find and graph the linear function that is the best fit for the U.S./Canada data.
c. Let x = the number of years after 2000 and draw a scatter plot of the international data.
d. Find and graph the linear function that is the best fit for the international data.
e. If the models are accurate, will U.S./Canada box office revenue ever reach the level of international box office revenue?
44. U.S. Households with Internet Access The following table gives the percentage of U.S. households with Internet access in various years.
(Source: U.S. Census Bureau)
Year 1996 1997 1998 1999 2000
Percent 8.5 14.3 26.2 28.6 41.5
Year 2001 2003 2007 2008
Percent 50.5 52.4 61.7 78.0
a. Create a scatter plot of the data, with x equal to the number of years from 1995.
b. Create a linear equation that models the data. c. Graph the function and the data on the same
graph, to see how well the function models (fits) the data.
45. Marriage Rate The marriage rate per 1000 popula- tion for selected years from 1991 to 2009 is shown in the table.
Figure 2.25
(Source: Data from IHS Global Insight, in the Wall Street Journal, June 2010)
U.S. China 0 2.0 1.5 1.0 0.5 ’00
Note: Figures starting in 2010 are forecasts
’01 ’02 ’03 ’04 ’05 ’06 ’07 ’08 ’09 ’10 ’11 ’12 ’13 ’14
Size of the manufacturing sector (trillions of 2005 U.S. dollars) (Source: National Vital Statistics Report 2010)
Year Marriage Rate per 1000 Population Year Marriage Rate per 1000 Population 1991 9.4 2000 8.5 1992 9.3 2001 8.2 1993 9.0 2002 7.9 1994 9.1 2003 7.9 1995 8.9 2004 7.6 1996 8.8 2005 7.5 1997 8.9 2007 7.3 1998 8.4 2008 7.1 1999 8.6 2009 6.8
a. Create a scatter plot of the data, where x is the number of years after 1990.
b. Create a linear function that models the data, with
x equal to the number of years after 1990.
c. Graph the function and the data on the same axes, to see how well the function models the data. d. In what year is the marriage rate expected to be
6.5, according to the model?
2.3
Systems of Linear Equations
in Two Variables
KEY OBJECTIVES
■ Solve systems of linear equations graphically ■ Solve systems of linear
equations algebraically with the substitution method ■ Solve systems of linear
equations algebraically by elimination
■ Model systems of equa- tions to solve problems ■ Determine if a system of
linear equations is inconsis- tent or dependent
SECTION PREVIEW
China's ManufacturingFigure 2.25 shows that the size of the manufacturing sector of China will exceed that of the United States in this decade. If we find the linear functions that model these graphs, with x representing the number of years past 2000 and y representing the sizes of the manufacturing sector in trillions of 2005 dollars, the point of intersection of the graphs of these functions will represent the simultaneous solution of the two equations because both equations will be satisfied by the coordinates of the point. (See Example 5.)