5.5 writing linear equations

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5.5 writing linear equations

  1. 1. Parallel and Perpendicular Lines
  2. 2. <ul><li>If two non-vertical lines in the same plane have the same slope, then the lines are parallel . </li></ul><ul><li>If two non-vertical lines in the same plane are parallel , then they have the same slope. </li></ul><ul><li>Two horizontal lines in the same plane are parallel ; two vertical lines in the same plane are parallel . </li></ul>Parallel Lines: Concept
  3. 3. SOLUTION EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3 x – 1. STEP 1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3. STEP 2 Find the y - intercept. Use the slope and the given point. y = m x + b Write slope-intercept form .
  4. 4. EXAMPLE 1 Write an equation of a parallel line y = 3 x + b – 5 = 3 ( –3 ) + b 4 = b Substitute 3 for m . Substitute  3 for x , and  5 for y . Solve for b . STEP 3 Write an equation. Use y = mx + b . y = 3 x + 4 Substitute 3 for m and 4 for b . Using the point (–3, –5)
  5. 5. GUIDED PRACTICE for Example 1 <ul><li>Write an equation of the line that passes through </li></ul><ul><li>(–2, 11 ) and is parallel to the line y = –x + 5. </li></ul>y = –x + 9 ANSWER
  6. 6. Perpendicular Lines
  7. 7. <ul><li>If two non-vertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular . </li></ul><ul><li>If two non-vertical lines in the same plane are perpendicular , then their slopes are negative reciprocals. </li></ul><ul><li>A horizontal line and a vertical line in the same plane are perpendicular . </li></ul>Perpendicular Lines: Concept
  8. 8. EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which lines, if any, are parallel or perpendicular. Line a : y = 5 x – 3 Line b : x + 5 y = 2 Line c : –10 y – 2 x = 0 SOLUTION Find the slopes of the lines . <ul><li>Line a : The equation is in slope-intercept form . </li></ul><ul><li>The slope is 5. </li></ul>Write the equations for lines b and c in slope-intercept form.
  9. 9. EXAMPLE 2 5 y = – x + 2 – 10 y = 2 x Determine whether lines are parallel or perpendicular Line a : m = 5. Line b : x + 5 y = 2 Line c : – 10 y – 2 x = 0 y = – x 1 5 x y = 2 5 1 5 + –
  10. 10. EXAMPLE 2 ANSWER Determine whether lines are parallel or perpendicular <ul><li>Lines b and c have slopes of – , so they are parallel. </li></ul>1 5 <ul><li>Line a has a slope of 5, the negative reciprocal </li></ul><ul><li>of – , so it is perpendicular to lines b and c . </li></ul>1 5
  11. 11. GUIDED PRACTICE for Example 2 Determine which lines, if any, are parallel or perpendicular. Line a : 2 x + 6 y = –3 Line b : y = 3 x – 8 Line c : –1.5 y + 4.5 x = 6 ANSWER parallel: b and c ; perpendicular: a and b , a and c
  12. 12. SOLUTION EXAMPLE 3 Determine whether lines are perpendicular Line a : 12 y = –7 x + 42 Line b : 11 y = 16 x – 52 Find the slopes of the lines . Write the equations in slope-intercept form . The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they ? STATE FLAG
  13. 13. EXAMPLE 3 Determine whether lines are perpendicular Line a : 12 y = –7 x + 42 Line b : 11 y = 16 x – 52 ANSWER y = – x + 12 42 7 12 11 52 y = x – 16 11 The slope of line a is – . The slope of line b is . The two slopes are not negative reciprocals, so lines a and b are not perpendicular. 7 12 16 11
  14. 14. SOLUTION EXAMPLE 4 Write an equation of a perpendicular line Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2 x + 3. STEP 1 Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 –
  15. 15. EXAMPLE 4 STEP 2 Find the y - intercept. Use the slope and the given point. Write slope-intercept form. Solve for b . STEP 3 Write an equation . y = mx + b Write slope-intercept form . Write an equation of a perpendicular line – 5 = – ( 4 ) + b 1 2 Substitute – for m , 4 for x , and – 5 for y . 1 2 y = m x + b – 3 = b y = – x – 3 1 2 Substitute – for m and –3 for b . 1 2
  16. 16. GUIDED PRACTICE for Examples 3 and 4 3. Is line a perpendicular to line b ? Justify your answer using slopes. Line a : 2 y + x = –12 Line b : 2 y = 3 x – 8 ANSWER 1 No; the slope of line a is – , the slope of line b is . The slopes are not negative reciprocals so the lines are not perpendicular. 2 3 2
  17. 17. GUIDED PRACTICE for Examples 3 and 4 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4 x – 7 . ANSWER y = – x + 4 1 4
  18. 18. <ul><li>Parallel lines have the same slope. </li></ul><ul><ul><li>m 1 = m 2 </li></ul></ul><ul><li>The slopes of perpendicular lines are negative reciprocals of each other. </li></ul><ul><ul><li>m 1 . m 2 = -1 OR </li></ul></ul>Summary
  19. 19. Homework <ul><li>Exercise 5.5 page 322: 1-35, Odd. </li></ul>

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