8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation in two variables will have solutions written as ordered pairs. Unlike linear equations in a single variable, equations with two variables will, in general, have an infinite number of solutions.
8.2 Lines and Their Slopes 43 Graphing calculators can generate tables of ordered pairs. Here is an eample for 2 3 6. We must solve for to get Y 1 6 2X3 before generating the table. To find ordered pairs that satisf the equation, select an number for one of the variables, substitute it into the equation for that variable, and then solve for the other variable. For eample, suppose in the equation 2 3 6. Then 2 3 6 2 3 6 Let. 3 6 3 6 2, giving the ordered pair, 2. Other ordered pairs satisfing 2 3 6 include 6, 2, 3,, 3, 4, and 9, 4. The equation 2 3 6 is graphed b first plotting all the ordered pairs mentioned above. These are shown in Figure 1(a). The resulting points appear to lie on a straight line. If all the ordered pairs that satisf the equation 2 3 6 were graphed, the would form a straight line. In fact, the graph of an first-degree equation in two variables is a straight line. The graph of 2 3 6 is the line shown in Figure 1( b). 1 = (6 2)/3 1 1 1 ( 3, 4) 6 4 (, 2) 2 (3, ) 4 2 2 4 6 8 2 (6, 2) 4 (9, 4) 6 1 ( 3, 4) 2 + 3 = 6 (, 2) (3, ) (6, 2) (9, 4) 1 This is a calculator graph of the line shown in Figure 1. (a) FIGURE 1 Linear Equation in Two Variables An equation that can be written in the form A B C (A and B not both ) is a linear equation in two variables. This form is called standard form. All first-degree equations with two variables have straight-line graphs. Since a straight line is determined if an two different points on the line are known, finding two different points is enough to graph the line. Two points that are useful for graphing lines are the - and -intercepts. The -intercept is the point (if an) where the line crosses the -ais, and the -intercept is the point (if an) where the line crosses the -ais. (Note: In man tets, the
44 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities 4 = 3 or 1 = 4 + 3 1 intercepts are defined as numbers, and not points. However, in this book we will refer to intercepts as points.) Intercepts can be found as follows. 1 1 1 Intercepts To find the -intercept of the graph of a linear equation, let. To find the -intercept, let. The displa at the bottom of the screen supports the fact that 34, is the -intercept of the line in Figure 11. We could locate the -intercept similarl. ( 3_, ) 4 (, 3) 4 = 3 EXAMPLE 1 Find the - and -intercepts of 4 3, and graph the equation. To find the -intercept, let. To find the -intercept, let. 4 3 Let. 4 3 Let. 4 3 3 3 4 -intercept is 3,. 4 3 -intercept is, 3. The intercepts are the two points 34, and, 3. Use these two points to draw the graph, as shown in Figure 11. A line ma not have an -intercept, or it ma not have a -intercept. FIGURE 11 1 1 = 2 1 1 EXAMPLE 2 Graph each line. (a) 2 Writing 2 as 1 2 shows that an value of, including, gives 2, making the -intercept, 2. Since is alwas 2, there is no value of corresponding to, and so the graph has no -intercept. The graph, shown in Figure 12(a), is a horizontal line. 1 Compare this graph with the one in Figure 12(a). = 1 1 = 2 (, 2) Horizontal line = 1 ( 1, ) Vertical line 1 1 (a) 1 This vertical line is not an eample of a function (see Section 8.4), so we must use a draw command to obtain it. Compare with Figure 12. FIGURE 12 1 The form 1 1 shows that ever value of leads to 1, and so no value of makes. The graph, therefore, has no -intercept. The onl wa a straight line can have no -intercept is to be vertical, as shown in Figure 12.
8.2 Lines and Their Slopes 45 ( 2, 2 ) 2 1 ( 1, 1 ) ( 2, 1 ) 2 1 FIGURE 13 Slope Two different points determine a line. A line also can be determined b a point on the line and some measure of the steepness of the line. The measure of the steepness of a line is called the of the line. One wa to get a measure of the steepness of a line is to compare the vertical change in the line (the rise) to the horizontal change (the run) while moving along the line from one fied point to another. Suppose that 1, 1 and 2, 2 are two different points on a line. Then, going along the line from 1, 1 to 2, 2, the -value changes from 1 to 2, an amount equal to 2 1. As changes from 1 to 2, the value of changes from 1 to 2 b the amount 2 1. See Figure 13. The ratio of the change in to the change in is called the of the line. The letter m is used to denote the. Slope If 1 2, the of the line through the distinct points 1, 1 and 2, 2 is m rise change in run change in 2 1. 2 1 5 2 = 7 ( 5, 3) 3 ( 1) = 4 m= 4 4_ (2, 1) 7 = 7 FIGURE 14 EXAMPLE 3 Find the of the line that passes through the points 2, 1 and 5, 3. If 2,1 1, 1 and 5, 3 2, 2, then m 2 1 2 1 See Figure 14. On the other hand, if 2, 1 2, 2 and 5, 3 1, 1, the would be m 3 1 5 2 4 7 4 7. 1 3 4 2 5 7 4 7, the same answer. This eample suggests that the is the same no matter which point is considered first. Also, using similar triangles from geometr, it can be shown that the is the same no matter which two different points on the line are chosen. If we appl the formula to a vertical or a horizontal line, we find that either the numerator or denominator in the fraction is. EXAMPLE 4 Find the, if possible, of each of the following lines. (a) 3 B inspection, 3, 5 and 3, 4 are two points that satisf the equation 3. Use these two points to find the. m 4 5 9 3 3 Undefined Since division b zero is undefined, the is undefined. This is wh the definition of includes the restriction that. 1 2
46 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities 5 Find the b selecting two different points on the line, such as 3, 5 and 1, 5, and b using the definition of. m 5 5 3 1 4 Zero Highwa s are measured in percent. For eample, a of 8% means that the road gains 8 feet in altitude for each 1 feet that the road travels horizontall. Interstate highwas cannot eceed a of 6%. While this ma not seem like much of a, there are probabl stretches of interstate highwas that would be hard work for a distance runner. In Eample 2, 1 has a graph that is a vertical line, and 2 has a graph that is a horizontal line. Generalizing from those results and the results of Eample 4, we can make the following statements about vertical and horizontal lines. Vertical and Horizontal Lines A vertical line has an equation of the form a, where a is a real number, and its is undefined. A horizontal line has an equation of the form b, where b is a real number, and its is. If we know the of a line and a point contained on the line, then we can graph the line using the method shown in the net eample. EXAMPLE 5 Graph the line that has 23 and goes through the point 1, 4. First locate the point 1, 4 on a graph as shown in Figure 15. Then, from the definition of, change in m change in 2 3. Move up 2 units in the -direction and then 3 units to the right in the -direction to locate another point on the graph (labeled P). The line through 1, 4 and P is the required graph. Right 3 Up 2 ( 1, 4) P 4 2 1 2 4 FIGURE 15 The line graphed in Figure 14 has a negative, 47, and the line goes down from left to right. In contrast, the line graphed in Figure 15 has a positive
8.2 Lines and Their Slopes 47 Negative Zero Positive Undefined FIGURE 16, 23, and it goes up from left to right. These are particular cases of a general statement that can be made about s. (Figure 16 shows lines of positive, zero, negative, and undefined s.) Positive and Negative Slopes A line with a positive goes up (rises) from left to right, while a line with a negative goes down (falls) from left to right. Parallel and Perpendicular Lines The s of a pair of parallel or perpendicular lines are related in a special wa. The of a line measures the steepness of the line. Since parallel lines have equal steepness, their s also must be equal. Also, lines with the same are parallel. Slopes of Parallel Lines Two nonvertical lines with the same are parallel; two nonvertical parallel lines have the same. Furthermore, an two vertical lines are parallel. EXAMPLE 6 Are the lines L 1, through 2, 1 and 4, 5, and L 2, through 3, and, 2, parallel? The of L 1 is m 1 5 1 4 2 4 6 2 3. The of L 2 is m 2 2 3 Since the s are equal, the lines are parallel. 2 3 2 3. Perpendicular lines are lines that meet at right angles. It can be shown that the s of perpendicular lines have a product of 1, provided that neither line is vertical. For eample, if the of a line is 34, then an line perpendicular to it has 43, because 3443 1. Slopes of Perpendicular Lines If neither is vertical, two perpendicular lines have s that are negative reciprocals; that is, their product is 1. Also, two lines with s that are negative reciprocals are perpendicular. Ever vertical line is perpendicular to ever horizontal line.
48 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities EXAMPLE 7 Are the lines L 1, through, 3 and 2,, and L 2, through 3, and, 2, perpendicular? The of L 1 is 3 m 1 3 2 2. The of L 2 is m 2 Since the product of the s of the two lines is 3223 1, the lines are perpendicular. Average Rate of Change 2 3 2 3. We have seen how the of a line is the ratio of the change in (vertical change) to the change in (horizontal change). This idea can be etended to real-life situations as follows: the gives the average rate of change of per unit of change in, where the value of is dependent upon the value of. The net eample illustrates this idea of average rate of change. We assume a linear relationship between and. EXAMPLE 8 The graph in Figure 17 approimates the percent of U.S. households owning multiple personal computers in the ears 1997 21. Find the average rate of change in percent per ear for the ears 1998 to 21. Percent HOMES WITH MULTIPLE PCS 3 (21, 24.4) 25 2 15 1 (1998, 13.6) 5 1997 1998 1999 2 21 Year Source: The Yankee Group. FIGURE 17 To use the formula, we need two pairs of data. From the graph, if we let 1998, then 13.6 and if 21, then 24.4, so we have the ordered pairs 1998, 13.6 and 21, 24.4. B the formula, average rate of change 2 1 24.4 13.6 1.8 3.6. 2 1 21 1998 3 This means that the number of U.S. households owning multiple computers increased b an average of 3.6% each ear in the period from 1998 to 21.