Slope of a Line
The slope of a line in the
plane containing the x and y
axes is generally represented by the letter m
and defined as the change in the y
coordinate divided by the corresponding change in the x
coordinate, between two distinct points on the line. This is described
by the following equation:
This equation may be viewed
pictorially:
Characteristics of slopes include:
The concept of slopes is useful regarding parallel and perpendicular lines. Suppose L1 and L2 represent two nonvertical lines with slopes m1 and m2 respectively, then:
Example #1:
Solution #1:
Example #2:
Solution #2:
Given two points (x1, y1) and (x2, y2), the change in x from one to the other is x2 − x1, while the change in y is y2 − y1. Substituting both quantities into the above equation obtains the following:
Characteristics of slopes include:
- For horizontal lines, m = 0
- For lines rising from left to right, m > 0
- For lines falling from left to right, m < 0
- Vertical lines have no slope
The concept of slopes is useful regarding parallel and perpendicular lines. Suppose L1 and L2 represent two nonvertical lines with slopes m1 and m2 respectively, then:
L1 and L2 are parallel
Û m1 = m2
L1 and L2 are perpendicular
Û m1 m2 = −1
Example #1:
Find the slope of the line containing the points (−2, 5) and (3, 13)
Solution #1:
(Alternatively, slope could have been computed using (3,13) as (x1, y1) and (-2,5) as (x2,y2))
Example #2:
Find the slope of the line passing through (3, 13) that is perpendicular to the line containing the points (−2, 5) and (3, 13).
Solution #2: