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What Is the Slope between Two Points? (Mathematics Lesson)

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. It is important to remember that (x1, y1) is the point on the left and (x2, y2) is the point on the right.

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



x1, y1, x2 and y2 are symbols that represent the x-coordinates and y-coordinates of the points. In real questions, the Cartesian coordinates will have numbers, for example (1,1) and (3,5).

How to Find the Slope between Two Points

Finding the slope between two points is easy.

An Example Question

Find the slope between the points with Cartesian coordinates (1, 1) and (3, 5).
Step 1
Start with the formula.
Slope = (y2 - y1)/(x2 - x1)
Don't forget: / means ÷

Step 2
Find x1, y1, x2 and y2 from the Cartesian coordinates given in the question.

In our example, the Cartesian coordinates of the points are (1, 1) and (3,5). They are represented in the formula by (x1, y1) and (x2, y2).
(x1, y1) = (1, 1) ∴ x1 = 1, y1 = 1.
(x2, y2) = (3, 5) ∴ x2 = 3, y2 = 5.
Step 3
Substitute x1, y1, x2 and y2 into the formula.
Slope = (5 - 1)/(3 - 1)
Slope = (4)/(2)
Slope = 4 ÷ 2
Slope = 2
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.

A Real Example of How to Find the Slope between Two Points

Finding the slope between two points is often useful to find the slope of a line.

The slider below gives an example of finding the slope of a line passing through two points:
Interactive Widget
Here is an interactive widget to help you learn about finding the slope between two points.
Interactive Test
  show
 


Here's a second test on finding the slope between points.
Here's a third test on finding the slope between points.
Note

Positive and Negative Slopes

A positive slope means the line slopes up and to the right:



A negative slope means the line slopes down and to the right:

Zero Slope

A line that goes straight across has zero slope:

Slope of 1

A slope of 1 is a 45° line going from bottom-left to top-right:

Fractional Slope

Slope can be a fraction, such as ½ and ¾.

An improper fraction is positive, but less than 1.

A slope of 1 gives a 45° line that splits the graph in 2. A fractional slope is less steep than this:



Any slope greater than 1 is steeper than this.