Calculating Slope

We can tell whether a line's slope is big or small, and whether the slope is positive or negative. But what if we want to compare two big slopes, or two small slopes? We need a more exact definition of slope.

Let's start by drawing a line and picking two points on the line. (There's nothing special about this line or these points -- we just want an example.)

Slope is defined as the change in the y-coordinates divided by the change in the x-coordinates. People often remember this definition as "rise/run." In this picture, the change in y-coordinates (rise) is red, and the change in x-coordinates (run) is blue:

Writing "change in x-coordinates" and "change in y-coordinates" many times is a lot of work, so let's use the Greek letter delta, , as an abbreviation for change. The traditional abbreviation for slope is m. Now we can write a formula for slope:

If we name our first point (x₁, y₁) and our second point (x₂, y₂), we can rewrite our formula to get rid of the delta:

We can use this formula to find the slope of our example line. Our first point was (1, 2), so x₁ = 1 and y₁ = 2. Similarly, x₂ = 2 and y₂ = 4, because our second point was (2, 4).

m = (y₂ - y₁) / (x₂ - x₁)
= (4 - 2) / (2 - 1)
= 2/1
= 2.

Now we know that the slope of our line is 2. You can see from the graph that the line moves up two spaces for every space that it moves to the right: