Solving for two variables in terms of two other variables

Question

a = (bc) / (bc + d(1-c))

I'd like the variables a and c on one side of the equation, and the variables b and d on the other. Is this possible?

Answer 1

Yes, it is. Start by multiplying both sides by the right side denominator so there are no fractions.

The multiply everything out so that there are no parenthesis. You're left now with 4 terms.

If you look closely, you'll see that each term either has a $d$ or a $b$, but not both.

Now add and subtract from both sides as necessary to put all the $d$ terms on one side, and all the $b$ terms on the other.

Factor the $d$ and $b$ out on each side. Then divide so as to get $d \over b$ on one side and an expression with just $a$'s and $c$'s on the other.