Slopes of Lines

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In mathematics, the slope or steepness of a line describes its steepness, incline, or grade. 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. Given two points (x₁,y₁) and (x₂,y₂) on a line, the slope m of the line is

$m=\frac{y_2-y_1}{x_2-x_1}.$

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:

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

(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:

$m = \frac{y_2 - y_1}{x_2 - x_1}.$

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.

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:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}.$

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

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

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

Algebra

If y is a linear equation of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form

$y = mx + b \,$

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.

If the slope m of a line and a point (x₀,y₀) on the line are both known, then the equation of the line can be found using the point-slope formula:

$y - y_0 = m(x - x_0).\!$