State whether or not this formula shows a direct or inverse variation.

Question

My high school algebra textbook asks in the introduction to functions chapter to state whether or not this formula shows a direct or inverse variation. $$\large\dfrac{y_1}{x_1}=\dfrac{x_2}{y_2}$$

My approach is if I set the LHS equal to $k$ I get the direct variation $kx_1=y_1$ but to get a direct variation on the RHS I would have to set it equal to $\dfrac{1}{k}$ This leads to the fact that $k$ can only be $\pm1$. Does that restriction mean that the formula still shows a direct variation?

Answer 1

Variation is defined in terms of two variables at a time, so as initially stated the question is arguably incomplete. Another way of putting this is that the answer is "yes"; the equation shows both direct and inverse variation, depending on which pair of variables is considered.

For example, the equation is equivalent to $x_1 x_2 = y_1 y_2$, which can also be written $x_1 = \frac{y_1 y_2}{x_2}$. This shows that $x_1$ varies directly with $y_1$ and $y_2$, but varies inversely with $x_2$. Likewise, we can rewrite the equation to show the following:

• $x_2$ varies directly with $y_1$ and $y_2$, inversely with $x_1$.
• $y_1$ varies directly with $x_1$ and $x_2$, inversely with $y_2$.
• $y_2$ varies directly with $x_1$ and $x_2$, inversely with $y_1$.

In short: The $x$ variables vary inversely with each other, and directly with the $y$ variables; and vice-versa. But if the question is made specific to a certain pair of variables, then there will be a single unambiguous answer.