The Slope of Lines


The slope of a line is a measure of how steep it is. We generally write slope as a fraction, where the top of the fraction corresponds to the rise of the line, and the bottom of the fraction corresponds to the run of the line. For example, a line with a slope of 10/7 will rise 10 units for every 7 units it runs side to side. Look at the following graph of this blue line:

Note the triangle formed with the blue line, the red line, and the grey line. The blue line is the graphed line, while the red and grey lines are only here to show the concept. The red line here shows that there is a run of 2 units while the grey line shows there is a run of 3 units. Since slope is rise over run, and there is a rise of 3 units for each run of 2 units, the slope of this line is 3/2.

Slope is positive if the line is oriented like /, while it is negative if the line is oriented like \. Since the above line is oriented like /, the slope is a positive 3/2. Now look at this graph:

We’ve added two red dots to give you points to look at. Over the part of the line marked off with the red dots, there is a rise of 5 units and a run of 3 units. The line is oriented like \, making it negative. Since it’s negative, and slope is rise over run, the slope of this line is -5/3.

If you’re given two points on a line, you can find the slope without being given a graph. The rise of the line will be the difference between the y-coordinates, while the run will be the difference between the x-coordinates. This gives us the following formula for slope, which we represent with the variable m:

m = \frac{y_2-y_1}{x_2-x_1} 

So for example, if we are given the two points (-2, 4) and (0, 3), then our formula would give us:

m = \frac{3-4}{0-(-2)} = \frac{-1}{2} 

And that’s all there is to know about slope.