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What Is the Slope of a Vertical Line?

“What is the slope of a vertical line?” is a common question when you start to analyze graphs and linear equations. To answer that, let’s go back to the basic definition.

Slope is defined as the steepness, incline, or gradient of a line. It is also defined as the change in y value ("rise") over the change in x value ("run"). The slope formula is:

what is the slope of a vertical line: Slope formula

Finding Slope

Normally in a linear equation, the slope of the line is most easily calculated by putting the equation of the line in slope-intercept form, or y=mx+b format (as opposed to standard form).

Here, the variable m represents the slope. You can have a positive slope or negative slope depending on its value.

The formula for a horizontal line (y = -1, for example), matches the slope-intercept form, just without an mx. That means it has no slope!

However, the formulas for vertical lines (x = 4, for example), cannot be put into slope-intercept form. We'll show you why this is a little later.

What Is the Slope of a Horizontal Line?

The slope of a horizontal line is 0 because the line does not rise at all. In other words, for any two points on the straight line, the change in y-value will always be 0.

what is the slope of a vertical line: Slope of a horizontal line graph

What Is the Slope of a Vertical Line?

The slope of a vertical line is undefined. That’s because, in a horizontal line, the change in the x-value will always be 0. You can figure this out by calculating the horizontal difference between the two x-coordinates. Remember the slope formula:

Slope formula

With a vertical line, this results in a bottom denominator of 0. However, dividing by 0 is something that doesn’t exist in math! Even though the difference in y-coordinate may change, dividing any number by the change in x-coordinate, 0, will result in an undefined slope.

Slope of a vertical line graph

Remember the Slope Formula

When graphing linear equations, remember that m, the slope, is calculated by finding the vertical change between two points divided by the horizontal change between those two points.

However, remember these two unique cases: Horizontal lines have a slope of 0 because the vertical change is 0. Vertical lines have an undefined slope because the horizontal change is 0 — you cannot divide a number by 0.

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