slopes of lines

1. About Slopes

2. Definition of Slope
About Slopes

3. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
(x1, y1)
(x2, y2)
About Slopes

4. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δxm =
(x1, y1)
(x2, y2)
About Slopes

5. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
About Slopes

6. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Geometry of Slope
(x1, y1)
(x2, y2)
About Slopes

7. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
About Slopes

8. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
About Slopes

9. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
About Slopes

10. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
About Slopes

11. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
About Slopes

12. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
About Slopes

13. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
geometric
meaning
About Slopes

14. Example A. Find the slope of each of the following lines.
About Slopes

15. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
About Slopes

16. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
About Slopes

17. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes

18. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
m =
Δy
Δx
=
0
7
= 0

19. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0

20. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0

21. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0

22. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0

23. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0

24. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0

25. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0

26. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0

27. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0

28. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
m =
Δy
Δx
=
7
0
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0 (UDF)

29. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
About Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
m =
Δy
Δx
=
7
0
Horizontal line
Slope = 0
Vertical line
Slope is UDF
Tilted line
Slope = 0
= 0 (UDF)

30. Lines that go through the
quadrants I and III have
positive slopes.
About Slopes

31. Lines that go through the
quadrants I and III have
positive slopes.
About Slopes
III
III IV

32. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
About Slopes
III
III IV

33. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
About Slopes
III
III IV
III
III IV

34. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
About Slopes
The formula for slopes requires geometric information,
i.e. the positions of two points on the line.
III
III IV
III
III IV

35. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
About Slopes
The formula for slopes requires geometric information,
i.e. the positions of two points on the line.
However, if a line is given by its equation instead, we may
determine the slope from the equation directly.
III
III IV
III
III IV

36. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
About Slopes

37. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
About Slopes

38. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes

39. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

40. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

41. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

42. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

43. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

44. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

45. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0).

46. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0). Use these points to draw
the line.
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

47. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
About Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0). Use these points to draw
the line.
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.

48. b. 0 = –2y + 6
About Slopes

49. b. 0 = –2y + 6 solve for y
About Slopes

50. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
About Slopes

51. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
About Slopes

52. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
About Slopes

53. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
About Slopes

54. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
About Slopes

55. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
About Slopes

56. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes

57. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be
isolated because there is no y.

58. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.

59. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6  x = 2.

60. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6  x = 2.
This is the vertical line x = 2.

61. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
About Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6  x = 2.
This is the vertical line x = 2.

62. Two Facts About Slopes
I. Parallel lines have the same slope.
About Slopes

63. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
About Slopes

64. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
About Slopes

65. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
About Slopes

66. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
About Slopes

67. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
About Slopes

68. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
About Slopes

69. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes

70. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?

71. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y

72. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y2
3

73. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y
Hence the slope of 3x = 2y + 4 is .
2
3
2
3

74. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
About Slopes
b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y
Hence the slope of 3x = 2y + 4 is .
So L has slope –2/3 since L is perpendicular to it.
2
3
2
3

75. Summary on Slopes
How to Find Slopes
I. If two points on the line are given, use the slope formula
II. If the equation of the line is given, solve for the y and get
slope intercept form y = mx + b, then the number m is
the slope.
Geometry of Slope
The slope of tilted lines are nonzero.
Lines with positive slopes connect quadrants I and III.
Lines with negative slopes connect quadrants II and IV.
Lines that have slopes with large absolute values are steep.
The slope of a horizontal line is 0.
A vertical lines does not have slope or that it’s UDF.
Parallel lines have the same slopes.
Perpendicular lines have the negative reciprocal slopes of
each other.
rise
run=m =
Δy
Δx
y2 – y1
x2 – x1
=

76. Exercise A. Identify the vertical and the horizontal lines by
inspection first. Find their slopes or if it’s undefined, state so.
Fine the slopes of the other ones by solving for the y.
1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. 0 = 8 – 2x 5. y = –x + 4 6. 2x/3 – 3 = 6/5
7. 2x = 6 – 2y 8. 4y/5 – 12 = 3x/4 9. 2x + 3y = 3
10. –6 = 3x – 2y 11. 3x + 2 = 4y + 3x 12. 5x/4 + 2y/3 = 2
Exercise B.
13–18. Select two points and estimate the slope of each line.
13. 14. 15.
About Slopes

77. 16. 17. 18.
Exercise C. Draw and find the slope of the line that passes
through the given two points. Identify the vertical line and the
horizontal lines by inspection first.
19. (0, –1), (–2, 1) 20. (1, –2), (–2, 0) 21. (1, –2), (–2, –1)
22. (3, –1), (3, 1) 23. (1, –2), (–2, 3) 24. (2, –1), (3, –1)
25. (4, –2), (–3, 1) 26. (4, –2), (4, 0) 27. (7, –2), (–2, –6)
28. (3/2, –1), (3/2, 1) 29. (3/2, –1), (1, –3/2)
30. (–5/2, –1/2), (1/2, 1) 31. (3/2, 1/3), (1/3, 1/3)
32. (–2/3, –1/4), (1/2, 2/3) 33. (3/4, –1/3), (1/3, 3/2)
About Slopes

78. Exercise D.
34. Identify which lines are parallel and which one are
perpendicular.
A. The line that passes through (0, 1), (1, –2)
D. 2x – 4y = 1
B. C.
E. The line that’s perpendicular to 3y = x
F. The line with the x–intercept at 3 and y intercept at 6.
Find the slope, if possible of each of the following lines.
35. The line passes with the x intercept at x = 2,
and y–intercept at y = –5.
About Slopes

79. 36. The equation of the line is 3x = –5y+7
37. The equation of the line is 0 = –5y+7
38. The equation of the line is 3x = 7
39. The line is parallel to 2y = 5 – 6x
40. the line is perpendicular to 2y = 5 – 6x
41. The line is parallel to the line in problem 30.
42. the line is perpendicular to line in problem 31.
43. The line is parallel to the line in problem 33.
44. the line is perpendicular to line in problem 34.
About Slopes
Find the slope, if possible of each of the following lines