The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is

Through differential calculus, one can calculate the slope of the tangent line to a curve at a point.

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade m of a road is related to its angle of incline θ by

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".)

Given two points (x1,y1) and (x2,y2), the change in x from one to the other is x2 − x1 (run), while the change in y is y2 − y1 (rise). Substituting both quantities into the above equation obtains the following:

Note that the way the points are chosen on the line and their order does not matter; the slope will be the same in each case. Other curves have "accelerating" slopes and one can use calculus to determine such slopes.

Suppose a line runs through two points: P(1,2) and Q(13,8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

The slope is .

As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is