Slope is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. 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. It is also always the same thing as how many rises in one run.

Using calculus, one can calculate the slope of the tangent to a curve at a point.

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Definition

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:

$m\; =\; \backslash frac\{\backslash Delta\; y\}\{\backslash Delta\; x\}.$

(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, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following:

$m\; =\; \backslash frac\{y\_2\; -\; y\_1\}\{x\_2\; -\; x\_1\}.$

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.

Examples

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:

$m\; =\; \backslash frac\{\backslash Delta\; y\}\{\backslash Delta\; x\}\; =\; \backslash frac\{y\_2\; -\; y\_1\}\{x\_2\; -\; x\_1\}\; =\; \backslash frac\{8\; -\; 2\}\{13\; -\; 1\}\; =\; \backslash frac\{6\}\{12\}\; =\; \backslash frac\{1\}\{2\}.$

The slope is $\backslash textstyle\backslash frac\{1\}\{2\}\; =\; 0.5\backslash ,$.

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

$m\; =\; \backslash frac\{\; 21\; -\; 15\}\{3\; -\; 4\}\; =\; \backslash frac\{6\}\{-1\}\; =\; -6.$

Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. A vertical line's slope is undefined meaning it has "no slope."

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

$m\; =\; \backslash tan\backslash ,\backslash theta$

and

$\backslash theta\; =\; \backslash arctan\backslash ,m$