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The Slope of a Line

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Definition - The Slope of a Line

The slope of a line is a measure of how quickly or slowly the line is rising. Another way to think of it would be that the slope measures the steepness of a line. A slope that's very small, i.e. close to zero, indicates a line that's rising very slowly where a large slope indicates that the line is very steep.

If I give you two points, $(x_1, y_1)$ and $(x_2, y_2)$ then you can calculate the slope of the line connecting them using the slope formula:

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

The applet on the right gives you a chance to see what various slopes look like. Enter a numeric value into the text book, click on the "Draw Line" button and the tool will draw a line with the slope you specified.

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Example 1

Find the slope of the line through the points (3, 10) and (4, 5).

This is a straight-forward application of the formula. I'm going to let the numbers in the (3, 10) point by the x1 and y1 values and the numbers in the (4, 5) point will be the x2 and y2. Substituting those numbers into the formula gives us:


Example 2

Find the slope of the line through the points (-2, 8) and (4, -6).

This is also straight-forward application of the formula but watch carefully how I deal with the negative signs.

$$m=\frac{y_1-y_2}{x_1-x_2}=\frac{8-(-6)}{-2-4}=\frac{8 + 6}{-6}=\frac{14}{-6}=-\frac{8}{3}$$

Video Lectures


The slope of a line is one of its two fundamental characteristics. In this lecture, we'll talk about what the slope is and turn that into a formula that you can use to calculate the slope for a line. (lecture slides)
If we think of the slope as the rate of change of a line, there's a lot of information we can get about its behavior just knowing that one numeric value. In this lecture, we'll work through some examples of how to use the slope formula and see how to interpret the values that we get from it. (lecture slides)

Dyanmic Practice - Finding the Slope of a Line


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