Lesson 1-2 Slopes of LinesObject. To find the slope of a line and to determine

whether two lines are parallel, perpendicular or neither

2 1

2 1

y ym

x x

The slope of a line is a number representing the steepness of the line relative to the x-axis. Let (x1,y1) and (x2,y2) be any two points on a line. The difference in y values gives the rise and the difference in x values gives the run. The slope is the ratio of the rise to the run.

(x2,y2)

(x1,y1)

y2 – y1

x2 – x1

The figures below show lines with a positive slope. Lines with positive slope rise to the right.

(2,0 )

( 8,3)

3 0 3 1

8 2 6 2m

(2,0 )

( 5,3)

3 01

5 2m

( 0,3)

( 6,0)

12

3 0

0 6m

The figures to the left show lines with a negative slope. Lines with negative slope rise to the left.

Slopes of Parallel Lines

Two non vertical lines are parallel if and only if they have the same slope.

122y x

142y x

12y x(0,0)

(0,2)

(0,4)

(0,5)

x

y

Some Important facts about the slope of a line.

Horizontal lines have a slope of 0 because y2 – y1 = 0 for all y.

( 5,4) ( 9,4)

y2 – y1 = 0 slope = 0

Vertical Lines

Vertical lines have a slope of 0 because x2 – x1 = 0 for all x.

(2, 5)

(2, 1)

x2 – x1 = 0 No slope

It should be apparent that no slope is quite different from having Zero slope.

Example 1

• What are the slope and y-intercept of 7x + 13y = 26

7 + 13 y = 26 13 y = -7x + 26

13 7 + 26 = 13 13

7 y = + 213

x

y x

x

7The slope is and the y-intercept is 2.13x

Slopes of Perpendicular Lines

Two lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is if the slopes are m1 and m2 then m1 = - 1 or m1 • m2 = -1 m2

34y x

43y x

Practice Exercises

Find the slope of the line joining the given points. 1.) (0,0),(5,7)___ (2.) (2,-6),(8,-3)_____Find the slope and the y-intercept of the linear

equation given3.) 4x + 3y =9 m=___,b=____(4.) 3y = 11x m=__,b=___Tell which are graphs of perpendicular or parallel Lines.5.)(a) 3y=5x- 5 (b) y= -3/5 x + 4 (c) 10y = -6x – 7 6.) (a) y= 5/2x – 8 (b) -15x + 6y – 10 =0 (c) 4x+10y = 15

Homework 1-18 pg.1119-33 odd pp. 12-13