1. Chapter 3 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. The Slope of a Line 1 1 3 3 2 2 3.33.3 Find the slope of a line given two points. Find the slope from the equation of a line. Use slopes to determine whether two lines are parallel, perpendicular, or neither.

3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.” One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line. The Slope of a Line Slide 3.3 - 3

4. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 3.3 - 4 Find the slope of a line given two points.

5. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y-values, which is the difference 6 − 1 = 5 units. The horizontal change, or run, is the change in the x-values, which is the difference 5 − 2 = 3 units. Find the slope of a line given two points. Slide 3.3 - 5 The slope is the ratio of the vertical change in y to the horizontal change in x. Count squares on the grid to find the change. Upward and rightward movements are positive. Downward and leftward movements are negative.

6. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding the Slope of a Line Solution: Slide 3.3 - 6 Find the slope of the line.

7. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The slope of a line can be found through two nonspecific points. This notation is called subscript notation, read x 1 as “x-sub-one” and x 2 as “x-sub-two”. The slope of a line is the same for any two points on the line. Find the slope of a line given two points. (cont’d) Slide 3.3 - 7 Traditionally, the letter m represents slope. The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is Moving along the line from the point (x 1, y 1 ) to the point (x 2, y 2 ), we see that y changes by y 2 − y 1 units. This is the vertical change (rise). Similarly, x changes by x 2 − x 1 units, which is the horizontal change (run). The slope of the line is the ratio of y 2 − y 1 to x 2 − x 1.

8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solution: Finding Slopes of Lines Slide 3.3 - 8 and yield the same slope. Make sure to start with the x- and y- values of the same point and subtract the x - and y -values of the other point. Find the slope of the line through (6, −8) and (−2,4).

9. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Positive and Negative Slopes A line with a positive slope rises (slants up) from left to right. A line with a negative slope falls (slants down) from left to right. Find the slope of a line given two points. (cont’d) Slide 3.3 - 9 Slopes of Horizontal and Vertical Lines Horizontal lines, with equations of the form y = k, have slope 0. Vertical lines, with equations of the form x = k, have undefined slopes.

10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Slide 3.3 - 10 Find the slope of the line through (2, 5) and (−1,5). Finding the Slope of a Horizontal Line

11. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Finding the Slope of a Vertical Line Slide 3.3 - 11 Find the slope of the line through (3, 1) and (3,−4). undefined slope

12. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Find the slope from the equation of a line. Slide 3.3 - 12

13. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Consider the equation y = −3x + 5. The slope of the line can be found by choosing two different points for value x and then solving for the corresponding values of y. We choose x = −2 and x = 4. Find the slope from the equation of a line. Slide 3.3 - 13 The ordered pairs are (−2,11) and (4, −7). Now we use the slope formula.

14. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the slope from the equation of a line. (cont’d) Slide 3.3 - 14 Step 1: Solve the equation for y. Step 2: The slope is given by the coefficient of x. The slope, −3 is found, which is the same number as the coefficient of x in the given equation y = −3x + 5. It can be shown that this always happens, as long as the equation is solved for y. This fact is used to find the slope of a line from its equation, by:

15. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solution: Finding Slopes from Equations Slide 3.3 - 15 Find the slope of the line 3x + 2y = 9.

16. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Use slopes to determine whether two lines are parallel, perpendicular, or neither. Slide 3.3 - 16

17. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes. Use slopes to determine whether two lines are parallel, perpendicular, or neither. Slide 3.3 - 17 Lines are perpendicular if they intersect at a 90° angle. The product of the slopes of two perpendicular lines, neither of which is vertical, is always −1. This means that the slopes of perpendicular lines are negative (or opposite) reciprocals—if one slope is the nonzero number a, the other is. The table to the right shows several examples.

18. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Deciding whether Two Lines Are Parallel or Perpendicular Slide 3.3 - 18 Solution: The product of their slopes is −1, so they are perpendicular Determine whether the pair of lines is parallel, perpendicular, or neither.