Note that if lines have the same slope and the same y-intercept, the lines are the same. ������������������������ Parallel Lines Two nonvertical lines y1 = m1 x + b1 and y2 = m2 x + b2 are parallel if they have the same slope (that is, m1 = m2) and different y-intercepts. Vertical lines of the form x = a1 and x = a2 are parallel if a1 ≠ a2. Two nonvertical lines are perpendicular to each other if the lines form a right angle (90°) at their point of intersection. For this to happen, the graph of one line must be increasing (have positive slope) and the other line must be decreasing (have negative slope). Also, the rise of one line must be the run of the other line and vice versa. Mathematically, this means that the slopes of the lines are negative reciprocals of each other. Consider the graphs of the equations y = 3 4 x + 3 and y=- 4 3 x + 1. One line rises 3 units vertically and runs 4 units horizontally, so its slope is m = 3 4 . The other line falls 4 units vertically and runs 3 units horizontally, so its slope is m=- 4 3 . –8 –4 4 8 Down 4 These lines have slopes that are negative reciprocals and are perpendicular to one another. ������������������������ Perpendicular Lines Two nonvertical lines y1 = m1 x + b1 and y2 = m2 x + b2 are perpendicular if their slopes are negative reciprocals—that is, if m1=- 1 m2 Any vertical line of the form x = a is perpendicular to any horizontal line y = b. Perpendicular lines may or may not have the same y-intercept. ������������ The negative reciprocal means that the signs must be different and the fractions must be reciprocals of each other. ���������������������� Determining if Two Lines Are Parallel or Perpendicular Step 1: Write each equation in slope-intercept form. Step 2: Find the slope of each line. Step 3: Compare the slopes of the lines. a. If the slopes are the same and the y-intercepts are different, then the lines are parallel. b. If the slopes are negative reciprocals of one another, then the lines are perpendicular. Right 4 Right 3 –8 x y Up 3 242 Chapter 3 Linear Equations in Two Variables
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