Carl Horowitz

**University of Michigan**

Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have **joint variation** or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.

Sometimes there are equations or circumstances where you actually have things that vary in more than one element so example of that is in joint variation where we actually have a direct variation with two variables.

Example we have here is distance = rate x time, pretty straight forward formula. If you think about it, if your rate increases your distance will increase or if your time increases your distance increases so actually distance is dependent on both rate and time and there's a direct correlation between all three of those. We still have our constant of variation that k that we typically throw on these problems but in this case it's actually a 1 so we don't actually need to put it in there okay.

The other kind of variation where we actually can depend on more than one thing is what we call combined variation where we can have direct and inverse variation in the same equation so this one right here isn't a applied formula like distance equals rate times time but it is a formula in which the variable y varies directly with x but also inversely with z so what we have is a combined variation with we include both direct and inverse variation.