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Slope is the measure of how much a line moves up or down related to how much it moves left to right.


In this image, the slope of the line is  \frac{y_2 - y_1}{x_2 - x_1} .

Parallel lines are those that have the same slope and do not touch. Examples include latitude lines.



Slope is the change in the vertical distance of a line on a coordinate plane over the change in horizontal difference. In other words, it is the “rise” over the “run” or the steepness of a line. Slope is usually represented by the symbol m like in the equation  y = mx + b , m the coefficient of x represents the slope of the line.

Slope is computed by measuring the change in vertical distance divided by the change in horizontal difference, i.e.:

\ m = \frac{\Delta y}{\Delta x} = \frac{rise}{run}

The Greek uppercase letter \Delta represents change, in this case change in the y-coordinates divided by the change in x-coordinates.

Positive Slope/ Negative Slope

If a line goes up from left to right, then the slope has to be positive. For example, a slope of ¾ would have a “rise” of 3, or go up 3; and a “run” of 4, or go right 4. Both numbers in the slope are either negative or positive in order to have a positive slope.

If a line goes down from left to right, then the slope has to be negative. For example, a slope of -3/4 would have a “rise” of -3, or go down 3; and a “run” of 4, or go right 4. Only one number in the slope can be negative for a line to have a negative slope.

Other Types of Slope

There are two special circumstances, no slope and slope of zero. A horizontal line has a slope of 0 and a vertical line has an undefined slope.

Horizontal lines have the form: \ y = a  ; where a is a constant, i.e.  a \in R
Vertical lines have the form: \ x = a  ; where as is a constant, i.e.  a \in R

Determining Slope[edit]

To determine the slope you need some information. This can include two (or more) coordinates, a parallel slope and a coordinate, a perpendicular slope and a coordinate, or the y-intercept and slope.

For completely horizontal lines, the difference in y coordinates between any two points is 0, so the slope m = 0, indicating no steepness in the line at all. If the line extends between right-upper (+,+) and left-lower ( -, -) directions, then the slope is positive. As the slope increases, the line becomes steeper until the line is almost vertical when the slope is very large. When the slope m = 1, the line is diagonal with an angle halfway between the x and y axes. If the line extends between left-upper (-,+) and right-lower (+, -) directions, then the slope is negative. As the slope changes from 0 to very negative numbers, the steepness in the opposite direction increases. Compare the slope ( m ) values in the following graph of functions y = 1 (where
m = 0), y = (1/2) x + 1, y = x + 1, y = 2 x, y = -(1/2) x + 1, y = -x + 1, and y = -2 x + 1. For all two-variable linear equations that can be converted to linear functions, the same calculation applies to slopes for those lines.

Various Linear Slopes.PNG

Finding It[edit]

For the most part finding slope when given information is a simple matter. Simply take the slope equation y=mx+b and replace the variable with whatever information you know, and solve.

Two Coordinates[edit]

To find the slope with two coordinates, you must first find the slope. Use the standard equation \frac{y_1-y_2}{x_1-x_2}. Put that into the equation as m, and replace x and y with x and y from one of the coordinates. Solve for b. Put that into the equation and your done.

Example: (1,4) (4,8)

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

 m = \frac{4-1}{4-8}

 m = \frac{3}{-4}

Plug that right in.

 y = \frac{3}{-4}x+b

 4 = \frac{3}{-4}(1)+b

 4 = \frac{3}{-4}+b

 4-\frac{3}{-4} = b

 \frac{-19}{4} = b

Put that in the equation and you're done.

 y = \frac{3}{-4}x + \frac{-19}{4}