When solving "Prove this" problems, is it wrong to begin by plugging in the given information?

Question

Suppose $Ae^{iax} + Be^{ibx} = Ce^{icx}$ for some nonzero constants $A, B, C, a, b, c$, and for all $x$. Prove that $a = b = c$ and $A + B = C$.

When given questions like this, my first instinct is to plug in the givens and try to see if I reach a true conclusion like $1=1$ or something. Is this the wrong way to approach such problems?

Note: I am not asking for someone to solve this, I am merely asking for advice on my approach to solving such problems where I'm asked to "prove" something.

For example, this is what I did for this problem, yet it feels unfulfilled, almost as if I have not guaranteed the proof. If my technique is not wrong here, then it is my way of thinking of when a mathematical thing is "fully proven".

$$Ae^{iax}+Be^{ibx}=Ce^{icx}$$ $$Ae^{iax}+Be^{iax}=Ce^{iax}$$ $$e^{iax}(A+B)=Ce^{iax}$$ $$e^{iax}(C)=Ce^{iax}$$ $$Ce^{iax}=Ce^{iax}$$

Answer 1

No, it is not a proof. You have assumed exactly what you're trying to prove. You used in your proof that $a=b=c$ and $A+B=C$, but that is exactly what you're trying to show.

Answer 2

Doing what you do does not constitute a proof of course, but sometimes it may create some insight as to what might be happening, which could give you some inspiration for how to prove what you need to prove.

For your specific example, for example, you do gain some insight as to how the $a$, $b$, and $c$ relate to each other, and same for the $A$, $B$, and $C$.

So, in general, I think it is a good idea, sure!

Answer 3

It's an excellent way to get an intuition for what's going on, and sometimes there's a way to get a proof out of it. The problem that may occur is that some of your steps may lose information - for example, you could be multiplying by zero, resulting in a true equation even if the equation you began with was false.

But the thing is, that's the only issue. That means that if your steps are reversible - that is, if they're equally correct running backwards as forwards - then you can get a proof by just doing that. For example, here's a proof of the example you gave:

$Ce^{iax} = Ce^{iax}$

$e^{iax}(C) = Ce^{iax}$

$e^{iax}(A + B) = Ce^{iax}$

$Ae^{iax} + Be^{iax} = Ce^{iax}$

$Ae^{iax} + Be^{ibx} = Ce^{icx}$

What this proof does is start from something true and use only valid steps to reach a conclusion - that makes it a genuine proof. Unfortunately, it's a proof of $Ae^{iax} + Be^{ibx} = Ce^{icx}$ given the statements $A + B = C$ and $a = b = c$, which isn't what you wanted. But this sort of approach may give you insight into the correct proof of the desired result.