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Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 1 2.5 Other Equations of Lines Point-Slope Form Parallel and Perpendicular Lines Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Point-Slope Form Suppose that a line of slope m passes through the point (x1, y1). For any other point (x, y) to lie on this line, we must have (x, y) y– y 1 (x1, y1) y y1 m. x x1 x– x 1 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 3 Point-Slope Form Any equation y – y1 = m(x – x1) is said to be written in point-slope form and has a graph that is a straight line. The slope of the line is m. The line passes through (x1, y1). Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4 Example Find an equation of the line passing through (1, 4) with slope –7. Solution Use the point-slope form equation: y – y1 = m(x – x1) y – 4 = –7(x – 1) Substituting Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 5 Parallel and Perpendicular Lines Two lines are parallel if they lie in the same plane and do not intersect no matter how far they are extended. Two lines are perpendicular if they intersect at a right angle. If one line is vertical and another is horizontal, they are perpendicular. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6 Slope and Parallel Lines Two lines are parallel if they have the same slope or if both lines are vertical. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Example Determine whether the line passing through (1, 2) and (7, 5) is parallel to the line given by 1 f ( x ) x 1. 2 Solution The slope of the line passing through (1,2) and (7,5) is given by 52 3 1 m . 7 1 6 2 Since the graph of f (x) = (½)x – 1 also has a slope of ½, the lines are parallel. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 8 Slope and Perpendicular Lines Two lines are perpendicular if the product of their slopes is –1 or if one line is vertical and the other is horizontal. Thus, if one line has slope m (m 0), the slope of a line perpendicular to it is –1/m. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Example Consider the line given by the equation 5y = 3x + 10. a) Find an equation for a parallel line passing through (0, 6). b) Find an equation for a perpendicular line passing through (0, 6). Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10 Solution Both parts (a) and (b) require find the slope of the line given by 5y = 3x + 10. Solve for y to find the slope-intercept form: 5 y 3x 10 3 y x2 5 The slope is 3/5. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11 a) The slope of any parallel line will be 3/5. The slope-intercept form yields: 3 y x 6. 5 b) The slope of any perpendicular line will be –5/3 (the opposite of the reciprocal of 3/5). The slope-intercept form yields: 5 y x 6. 3 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12