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 (x_{1}, y_{1}) and (x_{2}, y_{2}) 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.

Negative Slope

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 y_{1} - y_{2} = 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 x_{1} - x_{2} = 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.