## Slope |

In **slope** or **gradient** of a *direction* and the *steepness* of the line.^{[1]} Slope is often denoted by the letter *m*; there is no clear answer to the question why the letter *m* is used for slope, but it might be from the "m for multiple" in the equation of a straight line "y = mx + b" or "y = mx + c".^{[2]}

Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.

The *steepness*, incline, or grade of a line is measured by the *direction* of a

- A line is
**increasing**if it goes**up**from left to right. The slope is**positive**, i.e. . - A line is
**decreasing**if it goes**down**from left to right. The slope is**negative**, i.e. . - If a line is horizontal the slope is
**zero**. This is aconstant function . - If a line is vertical the slope is
*undefined*(see below).

The rise of a road between two points is the difference between the altitude of the road at those two points, say *y*_{1} and *y*_{2}, or in other words, the rise is (*y*_{2} − *y*_{1}) = Δ*y*. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (*x*_{2} − *x*_{1}) = Δ*x*. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.

In mathematical language, the slope *m* of the line is

The concept of slope applies directly to *m* of a line is related to its angle of incline *θ* by the

Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.

As a generalization of this practical description, the mathematics of

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

- definition
- algebra and geometry
- statistics
- slope of a road or railway
- calculus
- other generalizations
- see also
- references
- external links

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 Greek letter * delta*, Δ, is commonly used in mathematics to mean "difference" or "change".)

Given two points (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), the change in *x* from one to the other is *x*_{2} − *x*_{1} (*run*), while the change in *y* is *y*_{2} − *y*_{1} (*rise*). Substituting both quantities into the above equation generates the formula:

The formula fails for a vertical line, parallel to the *y* axis (see

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:

- .
- Since the slope is positive, the direction of the line is increasing. Since |m|<1, the incline is not very steep (incline <45°).

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

- Since the slope is negative, the direction of the line is decreasing. Since |m|>1, this decline is fairly steep (decline >45°).

Other Languages

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norsk: Stigningstall

oʻzbekcha/ўзбекча: Burchak koeffitsiyenti

polski: Nachylenie

português: Declive

română: Pantă

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slovenčina: Smernica priamky

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文言: 斜率

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