What is the Definition of Slope in Algebra?

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Slope is an important concept in algebra. Used in everything from basic graphing to more advanced concepts like linear regression, slope is one of the primary numbers in a linear formula. Slope indicates a line's direction on an ​x​/​y​ axis and also determines how steep that line appears.

TL;DR (Too Long; Didn't Read)

Slope is a measure of a line's rise (the distance it travels up or down the ​y​ axis) divided by its run (the distance it travels along the ​x​ axis) as measured from left to right. It can be positive (increasing upward) or negative (decreasing downward).

So What Is Slope?

Slope is a measure of the difference in position between two points on a line. If the line is plotted on a 2-dimensional graph, the slope represents how much the line moves along the x axis and the y axis between those two points. Though slope may appear as a whole number at times, it is technically a ratio of the x and y movement.

In the line equation

y = mx + b

the slope of the line is represented by ​m​. If a given line was

y = 3x + 2

the slope of the line would be 3. Since it is a ratio, it could also be represented as

\frac{3}{1}

Positive and Negative Slope

Slope represents the movement of a line from left to right, regardless of where the line is located on an x/y axis. A line is said to have positive slope if it increases along both the x and y axis as it moves from left to right. If the line decreases along the y axis as it moves from left to right, it is said to have a negative slope. A line that moves horizontally or vertically without any movement along the other axis has zero slope with vertical lines sometimes being said to have infinite slope.

An equation with positive slope would appear like

y = 2x + 5

An equation with negative slope would appear like

y = -3x + 2

When sketching lines on a graph, lines with positive slope move "up" when traveling left to right while those with negative slope move "down."

Calculating Slope

Slope is a measure of a line's rise (the amount it changes along the y axis) divided by its run (the amount it changes along the x axis). For a pair of points along the line, in this instance labeled ​(​x1, ​y1)​ and ​(​x2, ​y2)​, the slope is calculated with the following formula:

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

The result can be positive or negative. As an example, the line between points ​(3, 2)​ and ​(6, 4)​ would have a slope of

m = \frac{4 - 2}{6 - 3} = \frac{2}{3}

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