The document discusses parallel and perpendicular lines, explaining that parallel lines have the same slope while perpendicular lines have slopes that are negative reciprocals. It provides examples of finding the slope of parallel and perpendicular lines and writing equations of lines given properties like a point and slope. Formulas for slope-intercept and point-slope forms of linear equations are also presented.

1. Chapter 8
Systems of Equations and Inequalities

2. Section 8-1
Parallel and Perpendicular Lines

3. Essential Questions
✤ How do you determine if two lines are parallel or perpendicular?
✤ How do you write equations of parallel and perpendicular lines?
✤ Where you’ll see this:
✤ Sports, travel, safety

4. Vocabulary
1. Negative Reciprocals:

5. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1

6. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1
a b
and − are negative reciprocals
b a

7. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1
a b
and − are negative reciprocals
b a
2 3
and − are negative reciprocals
3 2

8. Parallel Lines

9. Parallel Lines
✤ Don’t intersect

10. Parallel Lines
✤ Don’t intersect
✤ Same slope

11. Parallel Lines
✤ Don’t intersect
✤ Same slope

12. Parallel Lines
✤ Don’t intersect
✤ Same slope
✤ If two lines are parallel, then they have the same slope

13. Parallel Lines
✤ Don’t intersect
✤ Same slope
✤ If two lines are parallel, then they have the same slope
✤ If two lines have the same slope, then they are parallel

14. Perpendicular Lines

15. Perpendicular Lines
✤ Intersect at 90 degree angles

16. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals

17. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals

18. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals
✤ If two lines are perpendicular, then the product of their slopes is -1

19. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals
✤ If two lines are perpendicular, then the product of their slopes is -1
✤ If the product of the slopes of two lines is -1, then they are
perpendicular

20. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
b. the parallel line?

21. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
1
m=−
4
b. the parallel line?

22. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
1
m=−
4
b. the parallel line?
m=4

23. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.

24. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12

25. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x

26. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12

27. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3

28. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4

29. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2

30. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

31. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

32. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

33. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3 y − 3 = 2x + 2
y = 2x + 4
m=2
(-1, 3)

34. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3 y − 3 = 2x + 2
y = 2x + 4 y = 2x + 5
m=2
(-1, 3)

35. Formulas
Slope-intercept:
Point-slope:

36. Formulas
Slope-intercept:
y = mx + b
Point-slope:

37. Formulas
Slope-intercept:
y = mx + b
Point-slope:
y − y1 = m(x − x1 )

38. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4

39. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m=
3

40. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m= (2, 7)
3

41. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m= (2, 7)
3
y − y1 = m(x − x1 )

42. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3
y − y1 = m(x − x1 )

43. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3 4 8
y−7= x−
y − y1 = m(x − x1 ) 3 3

44. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3 4 8
y−7= x−
y − y1 = m(x − x1 ) 3 3
4 13
y= x+
3 3

45. Quick Questions
1. What is true about the slopes of all horizontal lines?
2. If you knew the coordinates of four vertices of a quadrilateral, how
could you use slope to determine if the figure is a parallelogram?

46. Homework

47. Homework
p. 336 #1-39 odd
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