What Is the Slope Between Two Points?

What Is the Slope Between Two Points?

The slope (or gradient) between two points measures the steepness of the line joining the points.

The slope between two points can be found using the formula below:

In the formula, (x1, y1) and (x2, y2) are the Cartesian coordinates of the two points.

The image below shows what we mean by the slope between the two points:

Note: (x1, y1) is the point on the left and (x2, y2) is the point on the right.

How to Find the Slope Between Two Points

Finding the slope between two points is easy.

Question

What is the slope between the points (1, 1) and (3, 5)?

Step-by-Step:

1

Start with the formula.:

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

Don't forget: / means ÷

2

Find the Cartesian coordinates of the points. In our example:

  • The first point is (1, 1), so x1 = 1 and y1 = 1.

  • The second point is (3, 5), so x2 = 3 and y2 = 5.

3

Substitute x1, y1, x2 and y2 into the formula.

$$Slope = \frac{5 - 1}{3 - 1}$$ $$\:\:\:\:\:\:\:\:\:\:\:\: = \frac{4}{2}$$ $$\:\:\:\:\:\:\:\:\:\:\:\: = 4 \div 2$$ $$\:\:\:\:\:\:\:\:\:\:\:\: = 2$$

Answer:

The slope between the points (1, 1) and (3, 5) is 2.

How to Visualize the Slope between Two Points

The slope between the points (1, 1) and (3, 5) is 2.

By plotting the points, we can visualize what the slope means.

To get from one point to the other (going left to right), you can see in the image above that you have to go 4 up and 2 across.

The slope is simply how far up you go over how far across ("the change in y over the change in x" or "the rise over the run"). In this example it is 4/2 = 2.

Another way of see this is by noticing that for each square you go across, you go 2 up.

Slider

The slider below gives another example of finding the slope between two points

Open the slider in a new tab

See Also

What is slope? What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate? What is a fraction? What is an improper fraction? Calculate the slope ( interactive widget)