how to find slope from two points

Calculating the Slope

To calculate the slope of a line you need only two points from that line, (x 1, y 1) and (x 2, y 2).

There are three steps in calculating the slope of a straight line when you are not given its equation.

Step One: Identify two points on the line.
Step Two: Select one to be (x 1 , y 1 ) and the other to be (x 2 , y 2 ).
Step Three: Use the slope equation to calculate slope.

Take a moment to work through an example where we are given two points.

Example

Let's say that points (15, 8) and (10, 7) are on a straight line. What is the slope of this line?

Step One: Identify two points on the line.
In this example we are given two points, (15, 8) and (10, 7), on a straight line.
Step Two: Select one to be (x 1 , y 1 ) and the other to be (x 2 , y 2 ).
It doesn't matter which we choose, so let's take (15, 8) to be (x 2, y 2). Let's take the point (10, 7) to be the point (x 1, y 1).
Step Three: Use the equation to calculate slope.
Once we've completed step 2, we are ready to calculate the slope using the equation for a slope:

We said that it really doesn't matter which point we choose as (x 1, y 1) and the which to be (x 2, y 2). Let's show that this is true. Take the same two points (15, 8) and (10, 7), but this time we will calculate the slope using (15, 8) as (x 1, y 1) and (10, 7) as the point (x 2, y 2). Then substitute these into the equation for slope:

We get the same answer as before!

Often you will not be given the two points, but will need to identify two points from a graph. In this case the process is the same, the first step being to identify the points from the graph. Below is an example that begins with a graph.

Example

What is the slope of the line given in the graph?
The slope of this line is 2.

[detailed solution to example]

Now, take a moment to compare the two lines which are on the same graph.

Notice that the line with the greater slope is the steeper of the two. The greater the slope, the steeper the line. Keep in mind, you can only make this comparison between lines on a graph if: (1) both lines are drawn on the same set of axes, or (2) lines are drawn on different graphs (i.e., using different sets of axes) where both graphs have the same scale.

You are now ready to try a practice problem. If you have already completed the first practice problem for this unit you may wish to try the additional practice.