7 Write a Linear Equation to Model Real-World Data As seen in previous sections of this chapter, equations of lines are used to describe many kinds of real-world situations. We will look at an example in which we must fi nd the equation of a line given some data. EXAMPLE 8 Since 1998, sulfur dioxide emissions in the United States have been decreasing by about 1080.5 thousand tons per year. In 2000, approximately 16,636 thousand tons of the pollutant were released into the air. (Statistical Abstract of the United States) a) Write a linear equation to model this data. Let x represent the number of years after 1998, and let y represent the amount of sulfur dioxide (in thousands of tons) released into the air. b) How much sulfur dioxide was released into the air in 1998? in 2004? Solution a) Ask yourself, “What information is given in the problem?” i) “. . . emissions in the United States have been decreasing by about 1080.5 thousand tons per year” tells us the rate of change of emissions with respect to time. Therefore, this is the slope. It will be negative since emissions are decreasing. m 1080.5 ii) “In 2000, approximately 16,636 thousand tons . . . were released into the air” gives us a point on the line. Let x the number of years after 1998. The year 2000 corresponds to x 2. y amount of sulfur dioxide (in thousands of tons) released into the air. Then, 16,636 thousand tons corresponds to y 16,636. A point on the line is (2, 16,636). Now that we know the slope and a point on the line, we can write an equation of the line using the point-slope formula: y y1 m(x x1) Substitute 1080.5 for m. Substitute (2, 16,636) for (x1, y1). y y1 m(x x1) y 16,636 1080.5(x 2) Substitute 2 for x1 and 16,636 for y1. y 16,636 1080.5x 2161 Distribute. y 1080.5x 18,797 Add 16,636 to each side. The equation is y 1080.5x 18,797. b) To determine the amount of sulfur dioxide released into the air in 1998, let x 0 since x the number of years after 1998. y 1080.5(0) 18,797 Substitute x 0. y 18,797 180 CHAPTER 4 Linear Equations in Two Variables and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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