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5.3 Writing Linear Equations Given Two Points

5.3 Writing Linear Equations Given Two Points. Objective : Write an equation of a line given two points on the line. Type #1 (5.1) slope (m) = y – intercept (b) = Type #2 (5.2) slope (m) = y – intercept (b) = Type #3 (5.3) slope (m) = y – intercept (b) = . ???.

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5.3 Writing Linear Equations Given Two Points

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  1. 5.3 Writing Linear Equations Given Two Points Objective: Write an equation of a line given two points on the line. Type #1 (5.1) slope (m) = y – intercept (b) = Type #2 (5.2) slope (m) = y – intercept (b) = Type #3 (5.3) slope (m) = y – intercept (b) = ??? ??? ???

  2. Writing Equations – Type #3 Ex. 1 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! This is a Type #3 problem. First, when two points are given, the first thing in your mind is to calculate the slope! Plug the points into the slope formula. ??? ??? or

  3. Writing Equations – Type #3 Ex. 1 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (4, 2) because both numbers are positive. 2 = (4) + b ???

  4. Writing Equations – Type #3 2 = (4) + b Solve the equation for b 2 = + b To write an equation, you need two things: slope m = y – intercept b =

  5. Writing Equations – Type #3 Ex. 2 Write an equation of the line that goes through the points (2, 5) and (1, -2). To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! This is a Type #3 problem. First, when two points are given, the first thing in your mind is to calculate the slope! Plug the points into the slope formula. ??? ??? or

  6. Writing Equations – Type #3 Ex. 2 Write an equation of the line that goes through the points (2, 5) and (1, -2). To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (2, 5) because both numbers are positive. 5 = (2) + b ???

  7. Writing Equations – Type #3 5 = (2) + b Solve the equation for b 5 = + b To write an equation, you need two things: slope m = y – intercept b =

  8. Write an equation of the line that goes through the points (0, 1) and (1, 4). • y = 3x + 4 • y = 3x + 1 • y = -3x + 4 • y = -3x + 1

  9. Writing Equations – Type #3 Ex. 3 Write an equation of the line that goes through the points (-1, 6) and (4, 9). To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! This is a Type #3 problem. First, when two points are given, the first thing in your mind is to calculate the slope! Plug the points into the slope formula. ??? ??? or

  10. Writing Equations – Type #3 Ex. 3 Write an equation of the line that goes through the points (-1, 6) and (4, 9). To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (4, 2) because both numbers are positive. 6 = (-1) + b ???

  11. Writing Equations – Type #3 6 = (-1) + b Solve the equation for b 6 = + b To write an equation, you need two things: slope m = y – intercept b =

  12. Write an equation of the line that goes through the points (-2, 1) and (-3, 3). • y = 2x + 5 • y = 2x – 3 • y = -2x + 5 • y = -2x – 3

  13. Writing Equations – Type #3 Sometimes, you are given a graph, such as How can you write out the equation? To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! First, we pick up two nice points. Second, you have to find the slope. Plug the points into the slope formula or simply use “rise over run”. (-6, 1) (-4, -3) ??? ??? or

  14. Writing Equations – Type #3 ??? To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (-6, 1) 1 = (-6) + b (-6, 1) (-4, -3)

  15. Writing Equations – Type #3 1 = (-6) + b Solve the equation for b 1 = + b To write an equation, you need two things: slope m = y – intercept b =

  16. Writing Equations – Type #3 You Try This!! Write the equation for the given graph To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! First, we pick up two nice points. Second, you have to find the slope. Plug the points into the slope formula or simply use “rise over run”. (-3, 1) ??? ??? (2, -2) or

  17. Writing Equations – Type #3 ??? To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (-3, 1) 1 = (-3) + b (-3, 1) (2, -2)

  18. Writing Equations – Type #3 1 = (-3) + b Solve the equation for b 1 = + b To write an equation, you need two things: slope m = y – intercept b =

  19. Writing Equations – Type #3 To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b. Ex. 4 Find the slope and y-intercept. 1) y = 3x – 7 y = mx + b m = 3, b = -7

  20. Writing Equations – Type #3 Find the slope and y-intercept. 2) y = x y = mx + b y = x + 0 3) y = 5 y = mx + b y = 0x + 5 m = b = 0 m = 0 b = 5

  21. Writing Equations – Type #3 • When two lines are perpendicular, their slopes are opposite reciprocal. • Ex. 5. Check the two lines are perpendicular. • y = 3x + 4 and y = -3x -6 • and • and

  22. Writing Equations – Type #3 • When two lines are parallel, their slopes are equal. • When two lines are perpendicular, their slopes are opposite reciprocal. • Ex. 6. Write an equation of a line through (6, –1) that is • parallel to line y = –2x + 5 • perpendicular to line y = 3x – 2

  23. Writing Equations – Type #3 • Ex. 6. Write an equation of a line through (6, -1) that is • parallel to line y = –2x + 5 • To write an equation, you need m and b. • When two lines are parallel, their slopes are equal. The slope of the other line is –2. So m = –2. • You are also given a point. You should be able to find the y-intercept b. (Type #2 problem) • y = mx + b • –1 = –2(6) + b

  24. Writing Equations – Type #3 • Ex. 6. Write an equation of a line through (6, -1) that is • parallel to line y = –2x + 5 • –1 = –2(6) + b • –1 = –12 + b • +12 +12 • 11 = b • So the equation of the line is • y = –2x + 11

  25. Writing Equations – Type #3 Ex. 6. Write an equation of a line through (6, -1) that is (2) perpendicular to line y = 3x – 2 To write an equation, you need m and b. When two lines are perpendicular, their slopes are opposite reciprocal. The slope of the other line is 3. So m = . You are also given a point. You should be able to find the y-intercept b. (Type #2 problem) y = m x + b –1 = (6) + b

  26. Writing Equations – Type #3 • Ex. 6. Write an equation of a line through (6, -1) that is • parallel to line y = –2x + 5 • –1 = (6) + b • –1 = –2 + b • +2 +2 • 1 = b • So the equation of the line is

  27. Summary • In the Type #3 problem, neither slope nor y-intercept is directly provided. Only two points are given. • When you are given two points, the first thing in your mind is to find the slope by using the slope formula. • After you find out the slope, the Type #3 problem becomes to a Type #2 problem. Please see the top box on P. 286. • When you are given a graph of line, just find two grid(latice) points. Then follow the step 2 and step 3 above.

  28. Summary • 5. When the two lines are parallel, their slopes are equal. • 6. When the two lines are perpendicular, their slopes are OPPOSITE RECIPROCAL.

  29. Assignment P. 288 #’s 6 – 34 (even), 45 – 47

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