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Determine slope from two points

Slope is the constant rate of change of a linear relationship, found from two distinct points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) as y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1}, representing vertical change per unit horizontal change. The sign indicates whether the relationship increases or decreases, and the value can be interpreted from a graph as rise over run; a horizontal line has slope zero, while a vertical line has an undefined slope.

Detailed Explanation: Determine slope from two points

Slope tells you the vertical change for every 1 unit of horizontal change. For two points, use

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

where mm is the slope.

Example: Find the slope of the line through (2,3)(2,3) and (6,11)(6,11).

  1. Label the coordinates:

(x1,y1)=(2,3),(x2,y2)=(6,11) (x_1,y_1)=(2,3), \qquad (x_2,y_2)=(6,11)

  1. Substitute the values into the slope formula:

m=11−36−2 m=\frac{11-3}{6-2}

  1. Simplify:

m=84=2 m=\frac{8}{4}=2

The slope is

2\boxed{2}

This means that for every increase of 11 in xx, the value of yy increases by 22. The slope is positive because the line rises from left to right.

Be consistent with the order of the points: subtract the yy-values in the same order as the xx-values. A horizontal line has slope 00; a vertical line has an undefined slope because its horizontal change is 00.

Learn by doing: Determine slope from two points

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    Slope of a Line from Coordinates of Points

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