2. CHAPTER 3 Graphs of Liner Equations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 3.1Graphs and Applications of Linear Equations 3.2More with Graphing and Intercepts 3.3Slope and Applications 3.4Equations of Lines 3.5Graphing Using the Slope and the y-Intercept 3.6Parallel and Perpendicular Lines 3.7Graphing Inequalities in Two Variables

3. OBJECTIVES 3.6 Parallel and Perpendicular Lines Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aDetermine whether the graphs of two linear equations are parallel. bDetermine whether the graphs of two linear equations are perpendicular.

4. y = 2x + 5 y = 2x – 3 The graphs shown here are of the linear equations y = 2x + 5 and y = 2x – 3. The slope of each line is 2. The y-intercepts are (0, 5) and (0,–3). The lines do not intersect and are parallel. 3.6 Parallel and Perpendicular Lines a Determine whether the graphs of two linear equations are parallel. Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

5. Parallel non-vertical lines have the same slope, m 1 = m 2, and different y-intercepts, b 1  b 2. Parallel horizontal lines have equations y = p and y = q, where p  q. Parallel vertical lines have equations x = p and x = q, where p  q. 3.6 Parallel and Perpendicular Lines Parallel Lines Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

6. EXAMPLE Solution:Solve the second equation for y. 3.6 Parallel and Perpendicular Lines a Determine whether the graphs of two linear equations are parallel. ADetermine whether the graphs of the lines y = –2x – 3 and 8x + 4y = –6 are parallel. (continued) Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

7. EXAMPLE The slope of each line is –2 and the y- intercepts are different. The lines are parallel. 3.6 Parallel and Perpendicular Lines a Determine whether the graphs of two linear equations are parallel. ADetermine whether the graphs of the lines y = –2x – 3 and 8x + 4y = –6 are parallel. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

8. Perpendicular lines in a plane are lines that intersect at a right angle. The measure of a right angle is 90 degrees. y = 2x – 3 3.6 Parallel and Perpendicular Lines b Determine whether the graphs of two linear equations are perpendicular. Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The slopes of the lines are 2 and –1/2. Note that 2(– 1/2) = –1. That is, the product of the slopes is –1.

9. Two non-vertical lines are perpendicular if the product of their slopes is –1, m 1  m 2. (If one lines has slope m, the slope of the line perpendicular to it is –1/m.) The opposite reciprocal. If one equation in a pair of perpendicular lines is vertical, then the other is horizontal. The equations are of the form x = a and y = b. 3.6 Parallel and Perpendicular Lines Perpendicular Lines Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

10. EXAMPLE The slopes are 4 and –1/4. The product of the slopes is –1. The lines are perpendicular. Solution: Solve the second equation for y. 3.6 Parallel and Perpendicular Lines b Determine whether the graphs of two linear equations are perpendicular. BDetermine whether the graphs of the lines y = 4x + 1 and x + 4y = 4 are perpendicular. (continued) Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

11. EXAMPLE The slopes are 4 and –1/4. The product of the slopes is –1. The lines are perpendicular. 3.6 Parallel and Perpendicular Lines b Determine whether the graphs of two linear equations are perpendicular. BDetermine whether the graphs of the lines y = 4x + 1 and x + 4y = 4 are perpendicular. Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.