8. Equations of Lines and Linear Models Equations of Lines If the slope of a line and a particular point on the line are known, it is possible to find an equation of the line. Suppose that the slope of a line is m and 1, 1 is a particular point on the line. Let, be an other point on the line. Then, b the definition of slope, m 1. 1
8. Equations of Lines and Linear Models 41 Multipling both sides b the line. 1 gives the point-slope form of the equation of Point-Slope Form The equation of the line through 1, 1 with slope m is written in point-slope form as 1 m 1. Maria Gaetana Agnesi (1719 1799) did much of her mathematical work in coordinate geometr. She grew up in a scholarl atmosphere; her father was a mathematician on the facult at the Universit of Bologna. In a larger sense she was an heir to the long tradition of Italian mathematicians. Maria Agnesi was fluent in several languages b age 1, but she chose mathematics over literature. The curve shown below, called the witch of Agnesi, is studied in analtic geometr courses. a EXAMPLE 1 Find the standard form of an equation of the line with slope 1, passing through the point, 5. Use the point-slope form of the equation of a line, with 1, 1, 5 and m 1. 1 m 1 5 1 5 1 15 17 Let, m 1 1 5, 1. Multipl b. If two points on a line are known, it is possible to find an equation of the line. First, find the slope using the slope formula, and then use the slope with one of the given points in the point-slope form. = a + a EXAMPLE Find the standard form of an equation of the line passing through the points 4, and 5, 7. First find the slope, using the definition. m Either 4, or 5, 7 ma be used as 1, 1 in the point-slope form of the equation of the line. If 4, is used, then 4 1 and 1. 1 m 1 1 4 9 7 5 4 1 9 Let, m 1 1, 1 4. 9 1 9 4 9 1 4 9 7 1 4 1 9 1 Multipl b 9. Distributive propert
414 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities Suppose that the slope m of a line is known, and the -intercept of the line has coordinates, b. Then substituting into the point-slope form gives 1 m 1 b m 1, 1 b b m m b. Add b to both sides. This last result is known as the slope-intercept form of the equation of the line. Slope-Intercept Form The equation of a line with slope m and -intercept, b is written in slope-intercept form as m b. a a Slope -intercept is, b. The importance of the slope-intercept form of a linear equation cannot be overemphasized. First, ever linear equation (of a nonvertical line) has a unique (one and onl one) slope-intercept form. Second, in the net section we will stud linear functions, where the slope-intercept form is necessar in specifing such functions. EXAMPLE (a) Write the equation of the line described in Eample 1 in slope-intercept form. We determined the standard form of the equation of the line to be Solve for to obtain the slope-intercept form. 17 17. Subtract. 1 17 Multipl b 1. The slope is 1 and the -intercept is, 17. (b) Write the equation of the line described in Eample in slope-intercept form. 1 9 1 9 1 1 1 9 1 9 The slope is 19 and the -intercept is, 19. If the slope-intercept form of the equation of a line is known, the method of graphing described in Eample 5 of the previous section can be used to graph the line.
8. Equations of Lines and Linear Models 415 Down 1 = _ + (, ) Right FIGURE 18.1 4.7 4.7.1 = _ + (, 1) = _ + 7_ This screen gives support to the result in Eample 5(a). EXAMPLE 4 Graph the line with the equation. Since the equation is given in slope-intercept form, we can easil see that the slope is and the -intercept is,. Plot the point,, and then, using the rise over run interpretation of slope, move down units (because of the in the numerator of the slope) and to the right units (because of the in the denominator). We arrive at the point, 1. Plot the point, 1, and join the two points with a line, as shown in Figure 18. (We could also have interpreted as and obtained a different second point; however, the line would be the same.) As mentioned in the previous section, parallel lines have the same slope and perpendicular lines have slopes that are negative reciprocals of each other. EXAMPLE 5 Find the slope-intercept form of the equation of each line. (a) The line parallel to the graph of 6, passing through the point (4, 5) The slope of the line 6 can be found b solving for. 6 6 a ~ Slope Subtract. Divide b. 5 + = 6 = + 4 m = The slope is given b the coefficient of, so m. See the figure. The required equation of the line through 4, 5 and parallel to 6 must also have slope. To find this equation, use the point-slope form, with and m 1, 1 4, 5. 5 4, m 1 5, 1 4 5 4 ( 4, 5) 5 5 8 8 15 Distributive propert Add 5 15. + = 6 4 = + 7 7 Combine like terms. We did not clear fractions after the substitution step here because we want the equation in slope-intercept form that is, solved for. Both lines are shown in the figure.
416 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities (b) The line perpendicular to the graph of 6, passing through the point (4, 5) To be perpendicular to the line 6, a line must have a slope that is the negative reciprocal of, which is. Use the point (4, 5) and slope in the point-slope form to obtain the equation of the perpendicular line shown in the figure. 5 4, m 1 5, 1 4 5 4 5 6 Distributive propert ( 4, 5) 5 + = 6 = + 11 4 11 Add 5. A summar of the various forms of linear equations follows. Summar of Forms of Linear Equations A B C (Neither A nor B is.) a b m b 1 m( 1 ) Vertical line Undefined slope; -intercept is a,. Horizontal line Slope is ; -intercept is, b. Slope-intercept form Slope is m; -intercept is, b. Point-slope form Slope is m; Line passes through 1, 1. A Linear Model Earlier eamples and eercises gave equations that described real data. Now we are able to show how such equations can be found. The process of writing an equation to fit a graph is called curve-fitting. The following eample is an illustration of this concept for a straight line. The resulting equation is called a linear model. EXAMPLE 6 Retail spending (in billions of dollars) on prescription drugs in the U.S. is shown in the graph in Figure 19 on the net page. (a) Write an equation that models the data. The data shown in the bar graph increase linearl; that is, a straight line could be drawn through the tops of an two bars that will pass close to the tops of all
8. Equations of Lines and Linear Models 417 the bars. We can use the data and the point-slope form of the equation of a line to get an equation that models the relationship between ear and spending on prescription drugs. If we let 7 represent 1997, 8 represent 1998, and so on, the given data can be written as the ordered pairs 7, 79 and 11, 155. The slope of the line through these two points is m 155 79 11 7 76 4 19. Thus, retail spending on prescription drugs increased b about $19 billion per ear. Using this slope, one of the points, sa 7, 79, and the point-slope form, we obtain 1 m 1 79 19 7 79 19 1 19 54. Point-slope form 1, 1 7, 79; m 19 Distributive propert Slope-intercept form Thus, retail spending (in billions of dollars) on prescription drugs in the U.S. in ear can be approimated b the equation 19 54. RETAIL SPENDING ON PRESCRIPTION DRUGS Spending (in billions of dollars) 175 15 15 1 75 5 5 79 1997 1998 1999 1 Year Source: American Institute for Research Analsis of Scott-Levin data. FIGURE 19 155 (b) Use the equation from part (a) to predict retail spending on prescription drugs in the U.S. in 4. (Assume a constant rate of change here.) Since 7 represents 1997 and 4 is 7 ears after 1997, 14 represents 4. Substitute 14 for in the equation. 19 54 1914 54 1 According to the model, $1 billion will be spent on prescription drugs in 4.