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