The slope is an essential mathematical concept that lays the groundwork for more advanced topics down the line.

On the surface, it may seem confusing, just like most of the math that has to do with graphs. However, when you shake that first impression off, you’ll find it to be simple.

In this post, we cover the different ways with which you can find the slope of a line so that you can use the method that suits you the most.

The slope lets us know both the steepness and direction of a line. It’s a concept closely related to our everyday lives. Just by looking at a hill, for example, you can tell how steep it is, which has a bearing on things like the effort one will exert when climbing it.

Another example: imagine a golfer hitting the ball, and let’s assume that the ball will travel in a straight line. Through observing the slope of the ball’s course, you can guess if it’ll be able to pass over a certain obstacle or not.

See what you just did in these two scenarios? You predicted the direction and steepness of both stationary and moving objects! That’s exactly what the slope is all about.

Now, enough with the real-world examples, and let’s talk math!

There are two ways to do so. You’ll either have to draw a graph and use it to find the slope of a line, or you can memorize a little equation and use it to find the slope directly.

Drawing a Graph

Before discussing how to find the slope by looking at or drawing a graph, you first need to be aware of the rise and the run.

See, where there’s a slope, there’s change. What’s changing? The vertical and the horizontal values. In other words, the line goes up or down on the Y-axis and moves forward or backward on the X-axis.

Fittingly, the change in the vertical values is called the rise, whereas in the horizontal ones called the run. It follows, then, that the slope should have something to do with how much the line is rising and running on the graph, right?

Actually, it has everything to do with that: the slope equals the rise over the run. After getting that out of the way, let’s find the slope through two straightforward steps!

- Find the Rise and Run

First, you’ll either pick two different points on the line if it’s already graphed for you, or they’re going to give you the points and you’ll graph them. In any case, it won’t make a difference. Any two points on the line will do, as the slope is constant all the way through.

Remember that any point on a graph has coordinates: a Y value and an X value. They’re expressed in this form ‘(X, Y).’

The X value expresses where the point lies on the horizontal axis, while the Y value expresses its position relating to the vertical axis.

Now, say, for instance, that the two points we’re using have the coordinates of (3, 2) and (5, 6) respectively.

By looking at the graph, you can see that the value of the X-axis was 3 at the first point, and then became 5 in the second. This means the change in the horizontal axis, aka, the run is 5 - 3 = 2.

Similarly, the value of the Y-axis was 2 at the first point, and in the second one became 6. That means the change in the vertical axis, aka, the rise is 6 - 2 = 4.

- Work Out the Slope

As the slope equals the rise over the run, the slope of this line will be 4 over 2, which equals 2. You may have noticed that we began by the value of the second point in both the vertical and horizontal planes. That’s because the rise equals Y2 - Y1, and the run is X2 - X1.

However, we could’ve easily switched the points, calling whatever we like point 1, and the other one point 2. That’ll be okay, as long as you remain consistent with subtracting.

Meaning, you need to subtract in the same order. If you begin with point 1 when subtracting the Y values, you have to begin with the same point when dealing with the X values.

Using the Equation

All it takes is the coordinates of two points anywhere on the line, and you can find the slope without drawing or looking at any graph. So, let’s use the points from the previous example. Point 1 is (3, 2) and point 2 is (5, 6).

- Memorizing the Equation

The equation is already easy to memorize, and it’ll be all the easier when you find out where it comes from.

As you can recall, the slope is the rise over the run. We’ve also mentioned that the rise equals Y2 - Y1, and the run X2 - X1. This means the slope is Y2 - Y1 over X2 - X1, which gives us the following equation: m = Y2 - Y1 ÷ X2 - X1, where m is the slope. It’s that simple!

- Plugging in the Numbers

Without using any graph, you can just plug in the numbers from the coordinates into the equation, and voila; you’ve got your answer in one step!

In this case, m = 6 - 2 ÷ 5 - 3 = 4 ÷ 2 = 2. The same result as before!

Also, just like with the graph method, assigning any of the points as point 2 or 1 won’t make a difference in the result. As long as you’re following the same order when subtracting, all will be good.

Actually, let’s reverse the roles and designate (5, 6) as point 1 this time, and (3, 2) as point 2. Now, let’s put those newly arranged numbers in the equation: m = 2 - 6 ÷ 3 - 5 = -4 ÷ -2 = 2.

When a negative number is divided by another negative number, the result is positive. That’s why, even after rearranging the order of the points, the result remained unchanged.

Finding the slope of a line is a breeze, and once you understand it, it’ll help you weather many storms of future mathematical topics.

You just need to pay close attention to the order of subtraction after assigning your two points. This is a pitfall many students encounter.

Beyond that, whether you use a graph or the equation directly, simply tackle the question with confidence, and you’ll nail the answer!