Chapter 8
Systems of Equations and Inequalities
Section 8-1
Parallel and Perpendicular Lines
Essential Questions


✤   How do you determine if two lines are parallel or perpendicular?

✤   How do you write equations...
Vocabulary

1. Negative Reciprocals:
Vocabulary

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

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


               a            b
            ...
Vocabulary

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


               a            b
            ...
Parallel Lines
Parallel Lines


✤   Don’t intersect
Parallel Lines


✤   Don’t intersect

✤   Same slope
Parallel Lines


✤   Don’t intersect

✤   Same slope
Parallel Lines


✤   Don’t intersect

✤   Same slope



✤   If two lines are parallel, then they have the same slope
Parallel Lines


✤   Don’t intersect

✤   Same slope



✤   If two lines are parallel, then they have the same slope

✤   ...
Perpendicular Lines
Perpendicular Lines

✤   Intersect at 90 degree angles
Perpendicular Lines

✤   Intersect at 90 degree angles

✤   Slopes are negative reciprocals
Perpendicular Lines

✤   Intersect at 90 degree angles

✤   Slopes are negative reciprocals
Perpendicular Lines

✤   Intersect at 90 degree angles

✤   Slopes are negative reciprocals



✤   If two lines are perpen...
Perpendicular Lines

✤   Intersect at 90 degree angles

✤   Slopes are negative reciprocals



✤   If two lines are perpen...
Example 1
Suppose you have a line that has a slope of 4. What would be the slope
                                  of:

  ...
Example 1
Suppose you have a line that has a slope of 4. What would be the slope
                                  of:

  ...
Example 1
Suppose you have a line that has a slope of 4. What would be the slope
                                  of:

  ...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
                         to the line...
Formulas

           Slope-intercept:




            Point-slope:
Formulas

           Slope-intercept:

             y = mx + b


            Point-slope:
Formulas

           Slope-intercept:

              y = mx + b


             Point-slope:

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

       1. What is true about the slopes of all horizontal lines?




2. If you knew the coordinates of fo...
Homework
Homework



                          p. 336 #1-39 odd




“It is perhaps a more fortunate destiny to have a taste for col...
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Integrated Math 2 Section 8-1

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  • Integrated Math 2 Section 8-1

    1. 1. Chapter 8 Systems of Equations and Inequalities
    2. 2. Section 8-1 Parallel and Perpendicular Lines
    3. 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. 4. Vocabulary 1. Negative Reciprocals:
    5. 5. Vocabulary 1. Negative Reciprocals: Two rational numbers whose product is -1
    6. 6. Vocabulary 1. Negative Reciprocals: Two rational numbers whose product is -1 a b and − are negative reciprocals b a
    7. 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. 8. Parallel Lines
    9. 9. Parallel Lines ✤ Don’t intersect
    10. 10. Parallel Lines ✤ Don’t intersect ✤ Same slope
    11. 11. Parallel Lines ✤ Don’t intersect ✤ Same slope
    12. 12. Parallel Lines ✤ Don’t intersect ✤ Same slope ✤ If two lines are parallel, then they have the same slope
    13. 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. 14. Perpendicular Lines
    15. 15. Perpendicular Lines ✤ Intersect at 90 degree angles
    16. 16. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals
    17. 17. Perpendicular Lines ✤ Intersect at 90 degree angles ✤ Slopes are negative reciprocals
    18. 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. 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. 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. 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. 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. 23. Example 2 Write the equation for the line that passes through (-1, 3) and is parallel to the line 3y - 6x = 12.
    24. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 35. Formulas Slope-intercept: Point-slope:
    36. 36. Formulas Slope-intercept: y = mx + b Point-slope:
    37. 37. Formulas Slope-intercept: y = mx + b Point-slope: y − y1 = m(x − x1 )
    38. 38. Example 3 Write the equation for the line that passes through (2, 7) and is perpendicular to 3 y=− x+6 4
    39. 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. 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. 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. 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. 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. 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. 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. 46. Homework
    47. 47. Homework p. 336 #1-39 odd “It is perhaps a more fortunate destiny to have a taste for collecting shells than to be born a millionaire.” - Robert Louis Stevenson

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