Vertical Line: Equation & Slope

Instructor: Jason Furney

Jason has taught both College and High School Mathematics and holds a Master's Degree in Math Education.

In this lesson, we will study a vertical line and find the equation in which to represent it. We will explore the slope equation and use it to determine the slope of a vertical line.

Vertical Lines

When looking at linear equations on a graph we often see lines that go straight up and down and ones that go straight across (from left to right). These are two special linear equations. One is known as a vertical line and the other is known as a horizontal.


Today, we will discuss vertical lines. If you are finding it hard to remember which line is which, remember that horizontal lines (the ones that go from left to right) have the word horizon in them. The horizon is the line in the landscape we see when we look off into the distance. It too is horizontal. So if you see a line going up and down, it must be vertical!

Vertical lines are represented by a very specific type of equation. It doesn't quite hold to most linear equations you have seen. In the next section, we will explore the equation of a line, y = mx + b, and why the vertical line follows a different rule.

Finding the Equation of a Vertical Line

All lines need a slope, right? If that is true, then we should be able to find the slope using the equation.


Let's look at the following vertical line and try and find m.


According to the slope equation, we need to have two points. Luckily, we can choose any points on the line shown above. Let's use the points (6,0) and (6,2). Now, plug those into your slope equation and see what you get!


Once we do the calculations, we get the slope to be 2/0. But there is a problem here. Do you see it?

The slope in this case violates a huge rule in mathematics. We cannot divide by zero! Try any two points on any vertical line and you would run into the same problem. So you see, a vertical line does not have a slope!

If our line doesn't have a slope, then how do we write an equation? Take a look at the points from our example above. What two values are the same?

The answer, of course, is the sixes! In fact, pick any point on that line and I bet you are going to have a six in the x position. This is because the x-value determines where the line is located.

For our example, x = 6 describes the entirety of the line. As you can see by looking at it, no matter what the y-value is, x will always be six.

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