Either your Algebra I teacher, or your memory of your Algebra I teacher, is incorrect about what a “solution to an equation” is. A solution is a value of the variable—that is, a number that goes in place of the variable that makes the two sides of the equation equal to each other.
It is plainly obvious to me that 5x - 5x = 5 has no solution in the real numbers. Using Jasmine’s line of thinking, could we show this by examining every possible value of x and show there is no solution without invoking proof by contradiction? If I tried doing that I would start with all the “special” possible x’s such as 0, 1, 5 etc. and then look at all the other x’s. The problem might be when x approaches +/- infinity because infinity is weird.
That was Stoid
And you also stated that, to solve this, you’d have to say that 0=5. So what am I missing? I know what I’d say to someone who doesn’t say that (I’d send them to talk to you), or who says that zero does equal five (I’d send them to talk to you). But if you state that it can only be solved if zero equals five, and you add that zero doesn’t equal five, then, what, I just agree with you?
It may take me a couple of days to formulate a response to this.
So you are saying that Law of the Excluded Middle has “truthiness” because it is part of larger logical system that seems to work well when it is included and it is assumed to always be true? I can go along with that.
But what if we consider just the Law of the Excluded Middle all by itself without reference to anything else. Does the Law of the Excluded Middle have it’s own intrinsic basis for being true?
While you’re mulling it over, let me add this: what happens if we — so to speak — take math out of the equation for a moment?
Take, say, moral philosophy. Imagine someone is deciding whether to commit perjury, and asks me if it’s ever morally permissible to lie under oath. “Explain it to me like I’m five,” he adds.
I know what I’d say if I were trying to get him to “no.” I know what I’d say if I were trying to get him to “yes.” And I know that, either way, I might fail. Maybe he’ll agree with what I have to say, but maybe he won’t. I have my work cut out for me.
But now imagine he instead kicks things off by wagging a finger in my face and loudly insisting that it’s never morally permissible to lie under oath. And then, without breaking eye contact, he asks me whether it’s ever morally permissible to lie under oath.
I don’t know how to get that guy to “no.” As far as I can tell, he’s already at “no.” I don’t know what more to say to someone who just now told me that the answer is “no.” I could try to reason him into concluding that he’s made a mistake, and the answer is actually “yes,” but I can’t get closer to “no” if a guy (a) already concluded that the answer is “no,” and so (b) stated that the answer is “no.”
If you’re doing mathematics for its own sake, you can include or exclude any axioms that (dis)interest you. If you’re doing mathematics because you want to describe the world, the axioms you include or exclude affect the usefulness of your mathematics. Does the law of the excluded middle help or hinder what you’re using the math for? That is the basis for including or excluding it.
Funny how we can have both the Law of the Excluded Middle and the Fallacy of the Excluded Middle and yet both are useful depending on the context.
Because one is about binary and the other is about things which occur on a range of values.
The following is informal, but I think illustrative.
- Trying to leave Law of EM out of a discussion of Aristotelian / Boolean logic is trying to force-fit extra values into a fundamentally binary scenario. It’s fallacious in context.
- Trying to ignore Fallacy of EM while discussing situations which naturally involve ranges and shades of gray amounts to trying to force-fit binarity into a real-valued world. It’s fallacious in context.
So I’ll suggest the sensible (can’t quite say “logical”) thing to do is decide whether your context is inherently binary or not. If binary, LEM is a requirement and FEM is inapplicable / an error. If non-binary, then FEM is applicable and LEM is inapplicable / an error.
Viewed as above, there’s nothing terribly surprising about different rules or axioms for different situations.
And the key idea is NOT that true and false are magic values in philosphy about truthiness. It’s that in a binary equals two-valued world, there are only two values. Talking of a third is inherently nonsense. Like talking about negative gravity or conventional speeds faster than c. That way lies Treknobabble, not usable outcomes.
In that case, why are assuming it and not assuming it the only possible options?
Oh crap, It’s binary “why?” choices all the way down! 
I did sort of promise a response to this so my response is, “You should always agree with me.”
If any of you were pondering what inspired the OP, as I was myself, I think maybe it was my ALWAYS QUESTION AUTHORITY attitude.
In the OP I did restrict x to the real numbers. Now let us assume that x is infinity. Is the equation (5 x infinity) - (5 x infinity) = 5 a mathematical statement? If so, is it true or is it false?
ETA that in this equation x is the multiplication sign and not the variable x.
You’re missing it closer.
If you say that zero does equal five, I should disagree with you. But if you say of a given proposition ‘this is only true if zero equals five,’ and then you add that ‘zero doesn’t equal five,’ then it doesn’t really matter whether I agree with you or disagree with you; what matters is that you agree with you.
It is not “mathematical” for the normal sorts of math you could explain to a 15yo, much less a 5yo. So it’s truth or falsity within that normal math is undefined; the statement is just invalid, period.
I’ll leave it to the real mathematicians to expand on how it might be able to be considered “mathematical”.
Did I say that?
You said: “If we are asked to solve the equation for x we end up with the equation 0 = 5”
I think Jasmine’s specific post illustrates a difference between a mere polynomial (say), p(x), where x is a formal variable, and a polynomial equation p(x)=0. NB looking at the space of polynomials themselves I can assert things like x+x=2x or x-x=0, but that is not a priori meant to be an equation to be solved.
Is it? You said this was not over the real numbers, but you did not say what it was.
Think of a formal statement like you would actually program it into a computer to be proved or disproved; of course everything had better be defined.