Joint Variation at a Glance

Now, let's deal with joint variation, where one is never enough. In fact, one seriously depends on two others, making a mathematical trio. Turns out it's not so fancy-shmancy: joint variation is like direct variation, but it involves more than one variable. The general form looks like this:

y = kxz

...where y varies directly with x, and y also varies directly with z. In other words, y varies directly with the product of xz. The constant of variation is still our good buddy, k.

Typically, we use this formula to find the area of a rectangle:

A = lw

Here k = 1.

If either the length or the width increases, the area increases. If either the length or the width decreases, the area decreases. Just think of lots of moving parts, geometrically speaking.

Sample Problem

If y varies jointly with the product of x and z, and y = 72 when x = 4 and z = 9, find y when x = 5 and z = 6.

First, we bust out our general form for joint variation.

y = kxz

Plug in our values to find k.

72 = k(4)(9)

Solve for k.

72 = 36k

k = 2

Plug k back into our general form.

y = 2xz

Finally, find y when x = 5 and z = 6.

y = 2(5)(6)

y = 60

Sample Problem

If y varies jointly with the product of x and z, and y = 40 when x = 1 and z = 2, find y when x = 3 and z = 4.

You know the drill. Start with the general formula.

y = kxz

Plug in everything we know.

40 = k(1)(2)

Now we can find k.

40 = 2k

k = 20

Throw k back into our general equation.

y = 20xz

Last but definitely not least, find y when x = 3 and z = 4.

y = 20(3)(4)

y = 240

Sample Problem

If y varies jointly with the product of x and z, and y = 30 when x = 10 and z = 6, find y when x = 7 and z = 2.

Not to sound like a broken record here, but once again, we whip out the general formula.

y = kxz

Throw in our values for x, y, and z, then stir the pot and heat at 350°.

30 = k(10)(6)

Solve for k.

30 = 60k

k = 0.5

Plug k back into our general form.

y = 0.5xz

Demolish this problem by finding y when x = 7 and z = 2.

y = 0.5(7)(2)

y = 7