Solving for x when x is in the denominator of an equation I have the formula $$f(x) = \frac{xy}{x+y}$$ How do I get the equation in terms of x or y?
 A: Note: your question doesn't exactly make sense as written.  Here are two interpretations of it.

How to isolate $x$ in $z = \dfrac{xy}{x+y}$:
Let $z=\dfrac{xy}{x+y}$ where $x\ne -y$ (to avoid division by zero).


*

*Multiply both sides by $x+y$: $$z(x+y) = xy$$

*Distribute: $$xz + yz = xy$$

*Move all the terms with $x$ in them to the left and all the terms without $x$ to the right: $$xz - xy = -yz$$

*Factor out $x$: $$x(z-y) = -yz$$

*Assuming $z\ne y$ (to avoid division by zero), divide by $z-y$ on both sides: $$x = -\frac{yz}{z-y}\quad \Big(= \frac{yz}{y-z}\Big)$$



Rewrite the implicit functional equation $\frac{xy}{x+y} = c$, solving for $y$ as an explicit function of $x$:
Do the exact same steps as above, but solving for $y$ to get $$y = \frac{cx}{x-c}$$ for all $x\ne c$.
A: Let $z = f(x)$
$$z = \frac{xy}{x+y}$$
$$z(x+y) = xy$$
$$zx + zy= xy$$
$$zx = xy -zy$$
$$zx = y(x -z)$$
$$\frac{zx}{x-z} = y$$
By symmetry in the original equation, we can let $x \leftrightarrow y$ to get
$$\frac{zy}{y-z} = x$$
