# When manipulating equations, is it ALWAYS possible to reduce to a single instance of a variable on one side?

Forgive me, I'm terrible at remembering definitions and I haven't really been super into mathematics for a long time. Despite that, this is something I've always wondered about. Take this equation, for example: $$ab^2+2b=\frac{ac}{b}$$ You can get $$c$$ isolated on one side very easily, just multiply by $$\frac{b}{a}$$. $$\frac{b}{a}(ab^2+2b)=c$$ However, when doing the same for $$a$$ or $$b$$, for me at least, how to do so is not immediately apparent.

Which leads into my question: When manipulating an equation, is it ALWAYS possible to reduce an equation to a single instance of a variable on one side, no matter how many terms, variables, operations, instances of variables, functions, etc. are present within the original equation? If so, how can you prove it? And if not, why not?

Mind you, this is less of a question about specifics and more of one about a general rule of mathematics.

• "When manipulating an equation, is it ALWAYS possible to reduce an equation to a single instance of a variable on one side" No. "And if not, why not?" Why should it? – fleablood Nov 8 '18 at 0:40
• no, you can't always transform the original equation to something as simple, by example the variables $x$ and $y$ in the equation $\cos (xy)+2^x=y$ cannot be separated easily, and if you manage to separate it you surely get a more complicated expression than the original one (surely a very ugly series or something worse) – Masacroso Nov 8 '18 at 0:41
• One case when it's clear it's impossible is when $c$ is not a function of $a,b$. – user25959 Nov 8 '18 at 2:00