1. Math 51
3.3
The Slope of a Line

2. ________ is a measure of the _____________ of a line.

3. ________ is a measure of the _____________ of a line.
Slope

4. ________ is a measure of the _____________ of a line.
Slope
steepness

12. The calculation to find the slope of a line is: 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 (where 𝑥 1 , 𝑦 1 and ( 𝑥 2 , 𝑦 2 ) are two points on the line, and 𝑚 represents slope).

13. The calculation to find the slope of a line is: 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 (where 𝑥 1 , 𝑦 1 and ( 𝑥 2 , 𝑦 2 ) are two points on the line, and 𝑚 represents slope).

14. Ex 1. Find the slope of the line passing through (−3,4) and (−4,−2).

15. Ex 2.

16. Ex 2.

17. Ex 2.

18. Ex 2.
slope is positive

19. Ex 2.
slope is positive

20. Ex 2.
slope is positive

21. Ex 2.
slope is positive

22. Ex 2.
slope is positive

23. Ex 2.
slope is positive
slope is negative

24. Ex 2.

25. Ex 2.
slope is 0

26. Ex 2.
slope is 0
slope is undefined

27. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

28. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

29. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

30. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

31. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

32. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

33. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

34. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

35. The slope of the line 𝑦=2𝑥+1 is 2. Why
The slope of the line 𝑦=2𝑥+1 is 2. Why? Try plugging in 𝑥=0 and then 𝑥=1. 𝑥=0: 𝑦=2 0 +1=0+1=1 𝑥=1: 𝑦=2 1 +1=2+1=3 So, when 𝑥 increases by 1 (run), 𝑦 increases by 2 (rise). Thus, the slope is 2 1 =2.

36. In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation
In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation. Note that 𝑚 gives the slope, and 𝑏 gives the 𝑦-intercept.

37. In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation
In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation. Note that 𝑚 gives the slope, and 𝑏 gives the 𝑦-intercept.
slope-intercept form

38. In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation
In general, 𝒚=𝒎𝒙+𝒃 is called ____________________ of a linear equation. Note that 𝑚 gives the slope, and 𝑏 gives the 𝑦-intercept.
slope-intercept form

39. Ex 3. Find the slope of the line: 3𝑥+5𝑦=−1

40. Nonintersecting lines that lie in the same plane are called __________________________.

41. Nonintersecting lines that lie in the same plane are called __________________________.
parallel lines

42. What can you say about the slopes of parallel lines
What can you say about the slopes of parallel lines? If two lines have the same slope, are they guaranteed to be parallel?

43. What can you say about the slopes of parallel lines
What can you say about the slopes of parallel lines? If two lines have the same slope, are they guaranteed to be parallel?
They’re the same.

44. What can you say about the slopes of parallel lines
What can you say about the slopes of parallel lines? If two lines have the same slope, are they guaranteed to be parallel?
They’re the same.
Yes!

45. Lines that intersect at a right angle (90°) are called __________________________________.

46. Lines that intersect at a right angle (90°) are called __________________________________.
perpendicular lines

47. What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?

48. What is the slope of line m? What is the slope of line n?
𝟏 𝟑
What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?

49. What is the slope of line m? What is the slope of line n?
𝟏 𝟑
What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?
3 −1

50. What is the slope of line m? What is the slope of line n?
𝟏 𝟑
What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?
3 −1 =−𝟑

51. What is the slope of line m? What is the slope of line n?
𝟏 𝟑
What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?
3 −1 =−𝟑
1 3 −3 =−𝟏

52. What is the slope of line m? What is the slope of line n?
𝟏 𝟑
What is the slope of line m?
What is the slope of line n?
What happens when you multiply the slopes?
Slopes of perpendicular lines
are negative reciprocals
3 −1 =−𝟑
1 3 −3 =−𝟏

53. In general, product of slopes of two nonvertical perpendicular lines is ________. In other words, slopes of two nonvertical perpendicular lines are _____________________________. For example, if a line has slope 𝟐 𝟓 , then any perpendicular line will have slope ______.

54. In general, product of slopes of two nonvertical perpendicular lines is ________. In other words, slopes of two nonvertical perpendicular lines are _____________________________. For example, if a line has slope 𝟐 𝟓 , then any perpendicular line will have slope ______. −𝟏

55. In general, product of slopes of two nonvertical perpendicular lines is ________. In other words, slopes of two nonvertical perpendicular lines are _____________________________. For example, if a line has slope 𝟐 𝟓 , then any perpendicular line will have slope ______. −𝟏
negative reciprocals

56. In general, product of slopes of two nonvertical perpendicular lines is ________. In other words, slopes of two nonvertical perpendicular lines are _____________________________. For example, if a line has slope 𝟐 𝟓 , then any perpendicular line will have slope ______. −𝟏
negative reciprocals
− 𝟓 𝟐

57. Ex 4. Determine whether each pair of lines is parallel, perpendicular, or neither. 4𝑥−𝑦=4 8𝑥−2𝑦=−12 3𝑥−𝑦=4 𝑥+3𝑦=9 𝑦= 1 3 𝑥+1 𝑦=3𝑥−2

58. Ex 4. Determine whether each pair of lines is parallel, perpendicular, or neither. 4𝑥−𝑦=4 8𝑥−2𝑦=−12 3𝑥−𝑦=4 𝑥+3𝑦=9 𝑦= 1 3 𝑥+1 𝑦=3𝑥−2

59. Ex 4. Determine whether each pair of lines is parallel, perpendicular, or neither. 4𝑥−𝑦=4 8𝑥−2𝑦=−12 3𝑥−𝑦=4 𝑥+3𝑦=9 𝑦= 1 3 𝑥+1 𝑦=3𝑥−2

60. Summary
Slope: 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 (also, horizontal ↔ slope 𝟎; and vertical ↔ undefined slope) Slope-intercept form: 𝒚=𝒎𝒙+𝒃 parallel ↔ same slope (also two vertical lines are parallel) perpendicular ↔ slopes are negative reciprocals (also, vertical & horizontal lines are perpendicular)