Formula for Slope—Overview, Equation, Examples — Mashup Math (2024)

Your simple guide to the formula for slope and how to use it

When learning how to graph lines and linear functions—notably in y=mx+b form—and deal with points on the coordinate plane, it is incredibly important to understand slope and how to use the formula for slope to determine the slope of a line that passes through two known points.

This short guide on slope will cover the following topics:

  • What is the formula for slope?

  • What is the formula for a slope of a line?

  • Slope Formula Example

What is the formula for slope?

In math, the formula for slope is used to determine the steepness of a line that passes through two or more points. One of the most basic characteristics of slopes is that they can be positive (↗ increasing from left to right), negative (↘ decreasing from left to right), zero (↔ a horizontal line), or undefined (↕ a vertical line). The formula for slope is used to find the fraction that represents a ration between the change in y over the change in x. In other words, the formula for slope is the change in the y-coordinates over the change in the x-coordinates.

To find the slope of the lines that passes through the points (x1, y1) and (x2,y2), you can use the formula for slope as follows:

The formula for slope is…

m = Δy/Δx → m = (y2 - y1)/(x2 - x1)

In this formula:

  • m represents the slope of the line

  • Δy = change in y, Δx = change in x

  • x1 and x2 are the respective x-coordinates of the given points, while y1 and y2 are the respective y-coordinates of the given points

Note that the 1st point gets the 1 subscript (x1, y1) and the 2nd point gets the 2 subscript (x2, y2).

What is the formula for a slope of a line?

Now that you know that the formula for a slope, m, can be found using the equation m = (y2 - y1)/(x2 - x1), you can extend this understanding to finding the slope of a line.

We can express the equation of a line in slope-intercept form y=mx+b where m equals the slope of the line and b equals the y-intercept.

For example, a line with equation y = (1/3)x + 4 has a slope of 1/3 (m=1/3) and a y-intercept at 4 (b=4).

The formula for a slope of a line is the same formula for slope described above. As long as you know two points that a line passes through, you can use the formula for slope, m = (y2 - y1)/(x2 - x1), to find the slope of a line.

To prove that this is true, let’s take a look at the graph of the previously mentioned linear equation y=1/3x+4 and pick two points on the line that we can put into the formula for slope to see if our result is m=1/3.

The line with equation y=1/3x+4 passes through the points (3,5) and (9,7).

  • The first point (3,5) → (x1,y1)

  • The first point (9,7) → (x2,y2)

Using the formula for slope:

m = (y2 - y1)/(x2 - x1) m = (7 - 5)/(9 - 3) = 2/6 = 1/3

Since 2/6 can be simplified to 1/3, we have just concluded that the slope is indeed 1/3.

Slope Formula Example #1

Now that you understand the formula for slope, let’s work through a slope formula example where you have to find the slope of a line that passes through two given points.

Formula for Slope—Overview, Equation, Examples — Mashup Math (3)

In this first example, you have to find the slope of the line that passes through the points (2,4) and (4,3). To find the slope, simply identify the values of x1, x2, y1, and y2 and the plug these values into the formula for slope and solve (be sure to express your answer in reduced form if possible).

You can easily identify the values of x1, x2, y1, and y2 as follows:

  • The first point (2,4) → (x1,y1)

  • The first point (4,3) → (x2,y2)

Using the formula for slope:

m = (y2 - y1)/(x2 - x1) m = (3 - 4)/(4 - 2) = -1/2

Since -1/2 can not be simplified, we have just concluded that the slope of the line is -1/2.

Formula for Slope—Overview, Equation, Examples — Mashup Math (4)

Slope Formula Example #2

Are you starting to get the hang of using the slope formula? Let’s try another example.

For the second example, you have to find the slope of the line that passes through the points (1,-1) and (3,7).

You can figure out the slope by simply identifying of x1, x2, y1, and y2 and then using the formula for slope to solve. Again, be sure to express your answer in reduced form whenever possible.

You can label the values of x1, x2, y1, and y2 as follows:

  • The first point (1,-1) → (x1,y1)

  • The first point (3,7) → (x2,y2)

Using the formula for slope:

m = (y2 - y1)/(x2 - x1) m = (7 - -1)/(3- 1) = (7 + 1)/(3- 1) = 8/2

Notice that, in the denominator, (7- -1) became (7+1) since a double-negative becomes a positive.

Also notice that the end result of 8/2 is not one of the four choices. This is because 8/2 can be simplified as 4 (since 8 divided by 2 equals 4).

m = (y2 - y1)/(x2 - x1) m = (7 - -1)/(3- 1) = (7 + 1)/(3- 1) = 8/2 = 4

Now you can conclude that the slope of the line that passes through the points is 4.

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Do you want to learn more about slope?

Check out our free Intro to Slope lesson guide for students for more information, examples, and an animated video lesson.

And here are a few more resources you may find helpful

  • Parallel Slopes and Perpendicular Slopes

  • Graphing Linear Inequalities in 3 Easy Steps

  • Point-Slope Form Explained!

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Formula for Slope—Overview, Equation, Examples — Mashup Math (2024)

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