# Direct Variation Problem Solving

Let’s suppose you are comparing how fast you are driving (average speed) to how fast you get to your school.

Let’s suppose you are comparing how fast you are driving (average speed) to how fast you get to your school.

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(Note that you could also have an Indirect Square Variation or Inverse Square Variation, like we saw above for a Direct Variation.

This would be of the form $$\displaystyle y=\frac\texty=k$$.) Here is a sample graph for inverse or indirect variation.

We need to fill in the numbers from the problem, and solve for $$k$$. We can then cross multiply to get the new amount of money ($$y$$). If $$\displaystyle \begin\frac&=\frac\\\frac&=\frac\\\\y&=200\end$$ We can set up a proportion with the $$y$$’s on top (representing gallons), and the $$x$$’s on bottom (representing number of loads).

Remember that “per load” means “for See how similar these types of problems are to the Proportions problems we did earlier?

With direct variation, the $$y$$-intercept is always (zero); this is how it’s defined.

(Note that Part Variation (see below), or “varies partly” means that there is an extra fixed constant, so we’ll have an equation like $$y=mx b$$, which is our typical linear equation.) Direct variation problems are typically written: → $$\boldsymbol$$, where $$k$$ is the ratio of $$y$$ to $$x$$ (which is the same as the slope or rate).

Do you see how when the $$x$$ variable goes up, the $$y$$ goes down, and when you multiply the $$x$$ with the $$y$$, we always get the same number (Note that this is different than a negative slope, or negative $$k$$ value, since with a negative slope, we can’t multiply the $$x$$’s and $$y$$’s to get the same number).

So the formula for inverse or indirect variation is: → $$\displaystyle \boldsymbol$$ or $$\boldsymbol$$, where $$k$$ is always the same number.

(I’m assuming in these examples that direct variation is linear; sometime I see it where it’s not, like in a Direct Square Variation where $$y=k$$.

There is a word problem example of this here.) We can also set up direct variation problems in a ratio, as long as we have the same variable in either the top or bottom of the ratio, or on the same side. Don’t let this scare you; the subscripts just refer to the either the first set of variables $$(,)$$, or the second $$(,)$$.