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3.4 The Slope of a Line

Slope Slope of a Line The slope m of the line containing the points (x1, y1) and (x2, y2) is given by

Example Find the slope of the line through (0, 3 ) and (2, 5). Graph the line. If we let (x1, y1) be (0, 3) and (x2, y2) be (2, 5), then upward

Example Find the slope of the line containing the points (4, –3 ) and (2, 2). Graph the line. If we let (x1, y1) be (4, –3) and (x2, y2) be (2, 2), then downward

Helpful Hint When finding slope, it makes no difference which point is identified as (x1, y1) and which is identified as (x2, y2). Just remember that whatever y-value is first in the numerator, its corresponding x-value is first in the denominator.

Slope-Intercept Form When a linear equation in two variables in written in slope-intercept form, y = mx + b then m is the slope of the line and (0, b) is the y-intercept of the line. slope y-intercept is (0, b)

Example Find the slope and the y-intercept of the line Solve the equation for y. The slope of the line is 3/2 and the y-intercept is (0, 11/2).

Example Find the slope of the line x = 6. Vertical line Use two points (6, 0) and (6, 3). The slope is undefined.

Example Find the slope of the line y = 3. Horizontal line Use two points (0, 3) and (3, 3). The slope is zero.

Slopes of Vertical and Horizontal Lines The slope of any vertical line is undefined. The slope of any horizontal line is 0.

Appearance of Lines with Given Slopes x y Positive Slope Line goes up to the right Lines with positive slopes go upward as x increases. m > 0 Lines with negative slopes go downward as x increases. Negative Slope Line goes downward to the right x y m < 0

Appearance of Lines with Given Slopes x y Zero Slope horizontal line x y Undefined Slope vertical line

Parallel Lines Two nonvertical lines are parallel if they have the same slope and different y-intercepts. x y

Perpendicular Lines Two nonvertical lines are perpendicular if the product of their slopes is –1. x y

Parallel and Perpendicular Lines Example Are the following lines parallel, perpendicular, or neither. –5x + y = –6 x + 5y = 5 Find the slope of each line. The first equation has a slope of 5 and the second equation has a slope of 1/5, so the lines are perpendicular.