1. College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson

2. Equations and Inequalities 1

3. Lines 1.3

4. Lines In this section, we find equations for straight lines lying in a coordinate plane. The equations will depend on how the line is inclined. So, we begin by discussing the concept of slope.

5. The Slope of a Line

6. Slope of a Line We first need a way to measure the “steepness” of a line. This is how quickly it rises (or falls) as we move from left to right.

7. Slope of a Line We define ‘run’ to be the distance we move to the right and ‘rise’ to be the corresponding distance that the line rises (or falls). The slope of a line is the ratio of rise to run:

8. Slope of a Line The figure shows situations where slope is important. Carpenters use the term pitch for the slope of a roof or a staircase. The term grade is used for the slope of a road.

9. Run & Rise If a line lies in a coordinate plane, then: Run is the change in the x-coordinate between any two points on the line. Rise is the corresponding change in the y-coordinate.

10. Slope of a Line—Definition That gives us the following definition of slope. The slope m of a nonvertical line that passes through the points A(x 1, y 1 ) and B(x 2, y 2 ) is: The slope of a vertical line is not defined.

11. Slope of a Line The slope is independent of which two points are chosen on the line.

12. Slope of a Line We can see that this is true from the similar triangles in the figure:

13. Slope of a Line The figure shows several lines labeled with their slopes.

14. Positive & Negative Slopes Notice that: Lines with positive slope slant upward to the right. Lines with negative slope slant downward to the right.

15. Steep & Horizontal Lines The steepest lines are those for which the absolute value of the slope is the largest. A horizontal line has slope zero.

16. E.g. 1—Finding Slope of a Line Through Two Points Find the slope of the line that passes through the points P(2, 1) and Q(8, 5) Since any two different points determine a line, only one line passes through these two points.

17. E.g. 1—Finding Slope of a Line Through Two Points From the definition, the slope is: This says that, for every 3 units we move to the right, the line rises 2 units.

18. E.g. 1—Finding Slope of a Line Through Two Points The line is drawn here.

19. Point-Slope Form of the Equations of a Line

20. Equations of Lines Now let’s find the equation of the line that passes through a given point P(x 1, y 1 ) and has slope m.

21. Equations of Lines A point P(x, y) with x ≠ x 1 lies on this line if and only if: The slope of the line through P 1 and P is equal to m, that is,

22. Equations of Lines That equation can be rewritten in the form y – y 1 = m(x – x 1 ) Note that the equation is also satisfied when x = x 1 and y = y 1. Therefore, it is an equation of the given line.

23. Point-Slope Form of the Equation of a Line An equation of the line that passes through the point (x 1, y 1 ) and has slope m is: y – y 1 = m(x – x 1 )

24. E.g. 2—Equation of a Line with Given Point and Slope (a)Find an equation of the line through (1, –3 ) with slope –½. (b)Sketch the line.

25. E.g. 2—Equation of a Line Using the point-slope form with m = –½, x 1 = 1, y 1 = –3, we obtain an equation of the line as: Example (a)

26. E.g. 2—Equation of a Line The fact that the slope is –½ tells us that: When we move to the right 2 units, the line drops 1 unit. Example (b)

27. E.g. 2—Equation of a Line This enables us to sketch the line here. Example (b)

28. E.g. 3—Finding the Equation of a Line Through Two Given Points Find an equation of the line through the points (–1, 2) and (3, –4) The slope of the line is:

29. Using the point-slope form with x 1 = –1 and y 1 = 2, we obtain: E.g. 3—Finding the Equation of a Line Through Two Given Points

30. Slope-Intercept Form of the Equations of a Line

31. Equations of Lines Suppose a nonvertical line has slope m and y-intercept b. This means the line intersects the y-axis at the point (0, b).

32. Slope-Intercept Form of the Equation of a Line The point-slope form of the equation of the line, with x = 0 and y = b, becomes: y – b = m(x – 0) This simplifies to y = mx + b. This is called the slope-intercept form of the equation of a line.

33. Slope-Intercept Form of the Equation of a Line An equation of the line that has slope m and y-intercept b is: y = mx + b

34. E.g. 4—Lines in Slope-Intercept Form (a)Find the equation of the line with slope 3 and y-intercept –2. (b)Find the slope and y-intercept of the line 3y – 2x = 1.

35. E.g. 4—Slope-Intercept Form Since m = 3 and b = –2, from the slope- intercept form of the equation of a line, we get: y = 3x – 2 Example (a)

36. E.g. 4—Slope-Intercept Form We first write the equation in the form y = mx + b: From the slope-intercept form of the equation of a line, we see that: the slope is m = ⅔ and the y-intercept is b = ⅓. Example (b)

37. Vertical and Horizontal Lines

38. Equation of Horizontal Line If a line is horizontal, its slope is m = 0. So, its equation is: y = b where b is the y-intercept.

39. Equation of Vertical Line A vertical line does not have a slope. Still, we can write its equation as: x = a where a is the x-intercept. This is because the x-coordinate of every point on the line is a.

40. Vertical and Horizontal Lines An equation of the vertical line through (a, b) is: x = a An equation of the horizontal line through (a, b) is: y = b

41. E.g. 5—Vertical and Horizontal Lines (a)An equation for the vertical line through (3, 5) is x = 3. (b)The graph of the equation x = 3 is a vertical line with x-intercept 3.

42. E.g. 5—Vertical and Horizontal Lines (c) The equation for the horizontal line through (8, –2) is y = –2. (d) The graph of the equation y = –2 is a horizontal line with y-intercept –2.

43. General Equation of a Line

44. Linear Equation A linear equation is an equation of the form Ax + By + C = 0 where: A, B, and C are constants. A and B are not both 0.

45. Equation of a Line The equation of a line is a linear equation: A nonvertical line has the equation y = mx + b or –mx + y – b = 0 This is a linear equation with A = –m, B = 1, C = –b A vertical line has the equation x = a or x – a = 0 This is a linear equation with A = 1, B = 0, C = –a

46. Graph of a Linear Equation Conversely, the graph of a linear equation is a line: If B ≠ 0, the equation becomes: This is the slope-intercept form of the equation of a line (with m = –A/B and b = –C/B). If B = 0, the equation becomes: Ax + C = 0 or x = –C/A This represents a vertical line.

47. General Equation of a Line We have proved the following. The graph of every linear equation Ax + By + C = 0 (A, B not both zero) is a line. Conversely, every line is the graph of a linear equation.

48. E.g. 6—Graphing a Linear Equation Sketch the graph of the equation 2x – 3y – 12 = 0

49. E.g. 6—Graphing a Linear Equation Since the equation is linear, its graph is a line. To draw the graph, it is enough to find any two points on the line. The intercepts are the easiest points to find. Solution 1

50. E.g. 6—Graphing a Linear Equation x-intercept: Substitute y = 0. You get: 2x – 12 = 0. Thus, x = 6. y-intercept: Substitute x = 0. You get: –3y – 12 = 0. Thus, y = –4. Solution 1

51. E.g. 6—Graphing a Linear Equation With those points, we can sketch the graph as shown. Solution 1

52. E.g. 6—Graphing a Linear Equation We write the equation in slope-intercept form: This equation is in the form y = mx + b. So, the slope is m = ⅔ and the y-intercept is b = –4. Solution 2

53. E.g. 6—Graphing a Linear Equation To sketch the graph, we: 1.Plot the y-intercept. 2.Move 3 units to the right and 2 units up. Solution 2

54. Parallel and Perpendicular Lines

55. Parallel Lines Since slope measures the steepness of a line, it seems reasonable that parallel lines should have the same slope. In fact, we can prove this.

56. Parallel Lines Two nonvertical lines are parallel if and only if: They have the same slope.

57. Parallel Lines—Proof Let the lines l 1 and l 2 have slopes m 1 and m 2. If the lines are parallel, the right triangles ABC and DEF are similar. Thus,

58. Parallel Lines—Proof Conversely, if the slopes are equal, then the triangles will be similar. So, and the lines are parallel.

59. E.g. 7—Finding the Equation of a Line Parallel to a Given Line Find an equation of the line through the point (5, 2) that is parallel to the line 4x + 6y + 5 = 0.

60. First, we write the equation of the given line in slope-intercept form: So, the line has slope m = –⅔. Since the required line is parallel to the given line, it also has slope m = –⅔. E.g. 7—Finding the Equation of a Line Parallel to a Given Line

61. From the point-slope form of the equation of a line, we get: The equation of the required line is: 2x + 3y – 16 = 0 E.g. 7—Finding the Equation of a Line Parallel to a Given Line

62. Perpendicular Lines The condition for perpendicular lines is not as obvious as that for parallel lines.

63. Perpendicular Lines Two lines with slopes m 1 and m 2 are perpendicular if and only if m 1 m 2 = –1, that is, their slopes are negative reciprocals: Also, a horizontal line (slope 0) is perpendicular to a vertical line (no slope).

64. Perpendicular Lines—Proof Here, we show two lines intersecting at the origin. If the lines intersect at some other point, we consider lines parallel to these that intersect at the origin. These lines have the same slopes as the original lines.

65. Perpendicular Lines—Proof If the lines l 1 and l 2 have slopes m 1 and m 2, then their equations are: y = m 1 x and y = m 2 x Notice that: A(1, m 1 ) lies on l 1. B(1, m 2 ) lies on l 2.

66. Perpendicular Lines—Proof By the Pythagorean Theorem and its converse, OA ┴ OB if and only if:

67. Perpendicular Lines—Proof By the Distance Formula, this becomes:

68. E.g. 8—Perpendicular Lines Show that the points P(3, 3), Q(8, 17), R(11, 5) are the vertices of a right triangle.

69. E.g. 8—Perpendicular Lines The slopes of the lines containing PR and QR are, respectively, As m 1 m 2 = –1, these lines are perpendicular. So, PQR is a right triangle.

70. E.g. 8—Perpendicular Lines The triangle is sketched here.

71. E.g. 9— Finding an Equation of a Line Perpendicular to Given Line Find an equation of the line that is perpendicular to the line 4x + 6y + 5 =0 and passes through the origin. In Example 7, we found that the slope of the line 4x + 6y + 5 = 0 is –2/3. Thus, the slope of a perpendicular line is the negative reciprocal—that is, 3/2.

72. Since the required line passes through (0, 0), the point-slope form gives: y – 0 = 3/2(x – 0) y = 3/2x E.g. 9— Finding an Equation of a Line Perpendicular to Given Line

73. E.g. 10—Graphing a Family of Lines Use a graphing calculator to graph the family of lines y = 0.5x + b for b = –2, –1, 0, 1, 2. What property do the lines share?

74. E.g. 10—Graphing a Family of Lines The lines are graphed here in the viewing rectangle [–6, 6] by [–6, 6]. They all have the same slope. So, they are parallel.

75. Modeling with Linear Equations: Slope as Rate of Change

76. Rate of Change When a line is used to model the relationship between two quantities, the slope of the line is the rate of change of one quantity with respect to the other.

77. Rate of Change For example, this graph gives the amount of gas in a tank that is being filled. The slope between the indicated points is: The slope is the rate at which the tank is being filled—2 gallons per minute.

78. Rate of Change Here, the tank is being drained at the rate of 0.03 gallon per minute. The slope is –0.03.

79. Slope as Rate of Change The next two examples give other situations where the slope of a line is a rate of change.

80. E.g. 11—Slope as Rate of Change A dam is built on a river to create a reservoir. The water level w in the reservoir is given by w = 4.5t + 28 where: t is the number of years since the dam was constructed. w is measured in feet.

81. E.g. 11—Slope as Rate of Change (a)Sketch a graph of this equation. (b)What do the slope and w-intercept of this graph represent?

82. E.g. 11—Slope as Rate of Change This equation is linear. So, its graph is a line. Since two points determine a line, we plot two points that lie on the graph and draw a line through them. Example (a)

83. E.g. 11—Slope as Rate of Change When t = 0, w = 4.5(0) + 28 = 28 So, (0, 28) is on the line. When t = 2, w = 4.5(2) + 28 = 37 So, (2, 37) is on the line. Example (a)

84. E.g. 11—Slope as Rate of Change The line determined by those points is shown here. Example (a)

85. E.g. 11—Slope as Rate of Change The slope is m = 4.5. It represents the rate of change of water level with respect to time. This means that the water level increases 4.5 ft per year. Example (b)

86. E.g. 11—Slope as Rate of Change The w-intercept is 28, and occurs when t = 0. This represents the water level when the dam was constructed. Example (b)

87. E.g. 12—Fitting a Line to Data As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) in terms of the height h (in kilometers). Assume that the relationship between T and h is linear.

88. E.g. 12—Fitting a Line to Data (a) T he points appear to lie along a line? (b)Find the equation of the line that fits the data, and graph the line. (c) Use the equation you found in part (b) to estimate the temperature 2.5 km above the ground.

89. E.g. 12—Fitting a Line to Data (d) What does the slope of the line represent?

90. The points appear to lie along a line. The slope between the points (0,20) and (1,10) is – 10. The slope between (1,10) and (2,0) is also –10. Example (a) E.g. 12—Fitting a Line to Data

91. Let h represent the height, and let T represent the temperature. The slope of the line is –10, and the T-intercept is 20. The required expression is: T = –10h + 20 Example (b) E.g. 12—Fitting a Line to Data

92. At a height of h = 2.5 km, the temperature is: T = –10(2.5) + 20 = –25 + 20 = –5°C Example (c) E.g. 12—Fitting a Line to Data

93. The slope is: m = –10°C/km This represents the rate of change of temperature with respect to distance above the ground. So, the temperature decreases 10°C per kilometer of height. Example (d) E.g. 12—Fitting a Line to Data