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Slope (Mathematics Glossary)

What Is the Slope of a Line?

The slope of a line is its steepness.

The larger the slope, the steeper the line.

A line with a high slope is steeper than a line with a low slope

Understanding the Slope of a Line

The slope of a line is how far the line goes up (or down) divided by how far a line goes across (left to right).

Look at the line below:

The line goes up by 3 and across by 3

If we draw a triangle under the line and measure how far it goes up and across...

  • The line goes up by 3...

  • ...and across by 3.

Slope = How far up ÷ How far across
Slope = 3 ÷ 3
Slope = 1

The slope of the line is 1.

Real Examples of the Slope of a Line

  • Lines that slope from the bottom-left up to the top-right have a positive slope.

    positive slope of 2

    The line above has a slope of 2 because it goes up by 2 squares and across by 1.

  • Lines that slope from the top-left up to the bottom-right have a negative slope.

    negative slope of -2

    The line above has a slope of -2 because it goes down by 2 squares and across by 1.

  • Lines that do not slope up nor down have a slope of 0.

    zero slope
  • Lines that do not slope across have an undefined slope.

    undefined slope

Formulas to Find the Slope of a Line

We can find the slope of a line if we know how far it goes up and how far it goes across.

In Cartesian coordinates, the y-axis measures how far up a line is and the x-axis measures how far across a line is. This lets us define a formula to find the slope of a line:

slope equals change in y over change in x

Read more about finding the slope of a line

We can find the slope if we know two points (in Cartesian coordinates) on the line.

slope equals y 2 minus y 1 over x 2 minus x 1

Read more about finding the slope between two points

More About the Slope of a Line

The slider below gives more information about the slope of a line:

Interactive Test
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Here's a second test on slope.
Here's a third test on slope.

Note

Rise Over the Run

An easy way to remember the meaning of slope is "rise over run".

Rise is how far up (or down) a line goes.

Run is how far across a line goes.

rise over run

A Slope of 1

A slope of 1 gives a 45° line. It goes up and across by the same amount.

slope of 1

Fractional Slopes

A slope can be a fraction such as ½ or ¾.

slope of a half

A fractional slope is still positive (slopes from bottom-left to top-right) but it is less steep than a slope of 1.