In addition to its familiar meaning, the word "slope" has precise mathematical meaning. The slope of a line is the rise over the run, or the change in y divided by the change in x. To find the slope of a line, pick any two points on the line. Then subtract their x-coordinates and subtract their y-coordinates in the same order. Divide the difference of the y-coordinates by the difference of the x- coordinates:
Given two points (x1, y1) and (x2, y2) on a line, the slope of the line is equal to:
m = =
Example 1. Find the slope of the line which passes through the points (2, 5) and (0, 1):
m = = = 2. This means that every time x increases by 1 (anywhere on the line), y increase by 2, and whenever x decreases by 1, y decreases by 2.
If a line has a positive slope (i.e. m > 0), then y always increases when x increases and y always decreases when x decreases. Thus, the graph of the line starts at the bottom left and goes towards the top right.
Often, however, the slope of a line is negative. A negative slope implies that y always decreases when x increases and y always increases when x decreases. Here is an example of a graph with negative slope:
m = = = -
Thus, as x increases by 3, y decreases by 4, and as x decreases by 3, y increases by 4.
Horizontal and Vertical Lines
Sometimes, we will see equations whose graphs are horizontal lines. These are graphs in which y remains constant -- that is, in which y1 - y2 = 0 for any two points on the line:
m = = = 0.
The slope of any horizontal line is 0. In other words, as x increases or decreases, y does not change. x takes every possible value at a specific y value.
We will also see equations whose graphs are vertical lines. These are graphs in which x remains constant -- that is, in which x1 - x2 = 0 for any two points on the line:
m = = = undefined. We cannot divide a number by zero.
The slope of any vertical line is undefined.x does not increase or decrease; rather, y takes every possible value at a specific x value.