## 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 L

Example #1:

Solution #1:

Example #2:

Solution #2:

Given two points (*x*_{1},
*y*_{1}) and (*x*_{2},
*y*_{2}), the change in *x*
from one to the other is *x*_{2} − *x*_{1}, while the change in *y*
is *y*_{2} − *y*_{1}.
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 L

_{1}and L

_{2}represent two nonvertical lines with slopes m

_{1}and m

_{2 }respectively, then:

L

_{1}and L_{2 }are parallel Û m_{1}= m_{2}_{ }

L

_{1}and L_{2 }are perpendicular Û m_{1}m_{2}= −1Example #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 (x

_{1}, y_{1}) and (-2,5) as (x_{2},y_{2}))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: