Express a variable as a function of another

Question

If we have this formula :

$a/(1-$$1\over x$$-$$1\over y$$) = 1/(1-$$1\over xb$$-$$1\over yb$$)$

Is it possible to express $a$ as a function of $b$, independently of $x$ or $y$ ?

Thank you

Answer 1

We can transform the given equation \begin{align} a &= \frac{1-(1/x)-(1/y)}{1-(1/(xb))-(1/(yb))} \\ &= \frac{b}{b} \frac{1-(1/x)-(1/y)}{1-(1/(xb))-(1/(yb))} \\ &= \frac{b(1-(1/x)-(1/y))}{b-(1/x)-(1/y)} \\ &= \frac{b(1-(1/x)-(1/y))}{(b-1)+ 1-(1/x)-(1/y)} \\ &= \frac{b}{\frac{b-1}{1-(1/x)-(1/y)} + 1} \end{align} So only for $b=1$, the dependency on $x$ and $y$ vanishes, you end up with $a=1$ too, but for $b\ne 1$ the dependency is there.