# Algebra/Slope

Algebra
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## Slope

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

Algebra/Slope

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

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.

## Finding It

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

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.

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