Slope

Term used in mathematics.

The slope of a line is defined as "rise over run."

In the commonly used slope-intercept form for a line, y=mx+b, the slope is indicated by the constant m.

You can determine the slope between any two points in the Cartesian coordinate system by divding the difference of the y values by the difference in x values. For example, given points P(1,2) and Q(13,8) we can find the slope m.

           y₁ - y₂

      m = ————————

           x₁ - x₂

So,

            8 - 2     6      1

    PQ_m =   ——————— = ——— = ———

            13 -1     12     2

and we found that the slope is 1/2. Note that it is also equivalent if we switch the points:

            2 - 8     -6     1

    PQ_m =   ——————— = ——— = ———

            1 -13     -12    2

Given the slope of a line (y=mx+b) and the y intercept (where x=0), we can determine the equation of the line.

In our example with points P(1,2), Q(13,8) and slope 1/2, the y intercept is 3/2. So using our slope-intercept form for a line we have, y = 1/2x + 3/2. b is the y intercept, hence the name of this form, slope-intercept.

(Note: In this example we gave you the y intercept of 3/2 which you could have derived using the point-slope form of an equation for a line.)

A vertical line will have a slope of 1 (there's no run, that is x=0). A horizontal line will have a slope of 0 (there's no rise, y=0).

The angle the line makes with respect to the x axis can be used to calculate the slope. m = tan θ. (See trigonometry).

The concept of the slope of a line is fundamental to algebra, analytic geometry, trigonometry, and calculus to name a few.