Special case of linear equations: Vertical lines

What is the slope of a vertical line

When you're asked for the slope of a vertical line, it actually isn't too hard to logically work out. The slope tells us the rise over run of a line. In other words, it tells us the slant of a line. When you've got a line that is vertical, it continues upwards and downwards indefinitely. So what is its slant? The answer is that it is undefined. Why do vertical lines have an undefined slope? Let's take a look at the mathematical explanation through the line's equation.

Equation of a vertical line

The equation of a line has several forms, but let's take the slope intercept form for this explanation.

The slope-intercept equation looks like the following:

Do you recall what each component of the equation stands for? The m in this equation stands for the slope of the line, while the b stands for the y-intercept. How do we find m? We use m = rise/run! In a vertical line, all the points on the line has the same x coordinate. This means that we'll get a run of 0. Since in mathematics, if you divide by 0, you'll get the answer of "undefined", then the slope of a vertical line is also "undefined".

Now what does a vertical line's equation look like? Since the line is just vertically straight at a certain x coordinate and never changes, the equation is simply:

Let's try putting these concepts into use!

Example problems

Question 1:

Determine the line of equations from the following points.

i) (-5, 2), (-5, 3)

Solution:

Here we can see that the two given points plotted out shows us a vertical line. Its y-values changes, but its x-values don't. Therefore, to draw this line, what equation will you need? Remember from the above? The equation is x = ?, where the ? stands for the x-intercept of the line. Therefore, the equation = x=-5.

ii) (-3, 4), (-3, 8)

Solution:

Here we have -3 as the constant x-coordinates throughout the line. This means that in the end we'll get:

Equation = x=-3

iii) (b,n), (b,m)

Solution:

What happens when we don't have actual coordinates? You can still tell from these coordinates that the line is vertical due to both of the x-coordinates being "b". Since you've got that down, you now know that:

Equation = x=b

Question 2:

Write the equation of the line with the given information.

a) Vertical, passes through (3,5)

Solution:

In this question, we're expressedly told that the line we're working with is vertical. You won't need to determine this yourself. Since vertical line equations are simply the x-intercepts, we know that the "3" gives us all the info we'll need to solve this problem.

Equation = x=3

Want to take a look at different equations for vertical lines? Why not check out this online slider to see vertical lines that cross the x-intercept at different points.

Now that you've learned about vertical lines, want to proceed to upcoming chapters? Take a look at parallel and perpendicular lines in linear functions. Also look at what is a system of linear inequalities and how to graph linear inequalities in two variables, as well as graphing systems of linear inequalities.