Extraneous solutions to simple equations I had an interesting thought during my procrastination: is it legal to take an equation, say > $3 = a * b * c$ and do the following: > $3 = abc$ > $0 = abc - 3$ > $0 / a = bc - 3/a$ > $0 / b = c - 3/a/b$ > $0 / c = -3/a/b/c$ > $0 = -3/a/b/c$ But this is not true since no quotient (other than $0$ and $\infty$) equals precisely 0, therefore > $0 \not = -3/a/b/c$ My question is whether or not I did something wrong in the math. If so, please specify where and how. Otherwise, I'd like to know the following: 1) Is this extraneous solution caused by an assumption (such as when there's nothing on one side of the equation, it is assumed to be $0$)? 2) Is it possible to find extraneous solutions like this for any function? (If so, please give an example) 3) Any equation derived like this should still be equal to the original equation. Is there a reason why the steps are accurate (assuming they are), but the result isn't?

When you divide by $c$, you should find that

$$0 = 1 - 3/a/b/c$$

not

$$0 = 3/a/b/c$$

So there's no extraneous solution introduced.