Direct Variation at a Glance

As the song sort of goes, what goes up goes up, and what goes down goes down.

The general form for direct variation looks like this:

y = kx

This states that y varies directly with x, where k is a constant of variation. All that really means is that when x increases, y also increases. And as x decreases, y decreases.

Another way to say this is y is directly proportional to x. Or that y is in direct proportion to x.

Let's look an example of direct variation. If Marge is paid $8 an hour for babysitting eight pit bulls, then the amount per job depends on the number of hours she watches the snarling pups. We can turn this into an equation:

y = 8x

...where x represents the number of hours she worked and y represents how much money she'll make.

Sample Problem

A case of Red Bull Super X energy drinks has 24 bottles. How many bottles do 10 cases hold?

Let y be the number of bottles and x be the number of cases. The number of bottles varies directly with the number of cases.

Each case holds 24 bottles, so our equation is:

y = 24x

That means k, our constant of variation, is 24.

For our problem, x = 10 cases. Plugging that in gives us:

y = (24)(10)

Which finally gives us:

y = 240 bottles

Sample Problem

If y varies directly with x, and y = 20 when x = 4, find y when x = 10.

First, write the general form for direct variation:

y = kx

Plug in the given values for x and y, and solve for k.

20 = k(4)

k = 5

Now, plug k = 5 back into the general form

y = 5x

We can now find y when x = 10.

y = 5(10)

y = 50 

Sample Problem

The circumference of a circle (say, a circular firing squad) varies directly with its diameter. Write the equation to represent this variation and find the constant of variation.

Let y represent the circumference of the circle (er, squad) and x be the diameter.

y = kx

Comparing this to the known formula for the circumference of the circle C = πd shows us that in this case:

k = π