# Equations with No Solution **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [June 8, 2025, 3:14pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330 "2025-06-08T15:14:26Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [June 10, 2025, 1:53pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/61 "2025-06-10T13:53:08Z")
> [@Jasmine](#): > > So the equation has a solution: 5X - 5X = 0 > [@Jasmine](#): > > My Algebra I teacher would have said, "Your solution for 5x - 5x = ? is incorrect. Find the correct solution. He wouldn’t have apologized to me for giving me an equation that has no solution. 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. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 2:02pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/62 "2025-06-10T14:02:11Z")
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. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [June 10, 2025, 2:08pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/63 "2025-06-10T14:08:55Z")
> [@Saint\_Cad](#): > > Were you the one that started the math thread about frisbees and zero That was Stoid ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [June 10, 2025, 4:51pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/64 "2025-06-10T16:51:23Z")
> [@Ynnad](#): > > I stated that 0 = 5 is false in the second sentence of the OP. 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? ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 5:06pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/65 "2025-06-10T17:06:30Z")
> [@The\_Other\_Waldo\_Pepper](#): > > 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. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 5:20pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/66 "2025-06-10T17:20:44Z")
> [@Saint\_Cad](#): > > What we said was choosing a logical system that includes the Law of the Excluded middle (and not working with the field **Z** mod 5) the process is completely valid. 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? ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [June 10, 2025, 5:55pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/67 "2025-06-10T17:55:00Z")
> [@Ynnad](#): > > It may take me a couple of days to formulate a response to this 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.” ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [June 10, 2025, 6:06pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/68 "2025-06-10T18:06:18Z")
> [@Ynnad](#): > > 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? 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. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 6:49pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/69 "2025-06-10T18:49:33Z")
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. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [June 10, 2025, 7:24pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/70 "2025-06-10T19:24:01Z")
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. ---
**Author:** ![hogarth](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hogarth/32/1773_2.png) [@hogarth](https://boards.straightdope.com/u/hogarth)\ **Post date:** [June 11, 2025, 4:43am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/71 "2025-06-11T04:43:51Z")
> [@LSLGuy](#): > > It was “_why_ do we assume only true and false are possible?” In that case, why are assuming it and not assuming it the only possible options? ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [June 11, 2025, 11:17am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/72 "2025-06-11T11:17:57Z")
Oh crap, It’s binary “why?” choices all the way down! 😉 ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 11, 2025, 11:18am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/73 "2025-06-11T11:18:05Z")
> [@The\_Other\_Waldo\_Pepper](#): > > … then, what, I just agree with you? I did sort of promise a response to this so my response is, “You should always agree with me.” ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 11, 2025, 11:19am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/74 "2025-06-11T11:19:36Z")
If any of you were pondering what inspired the OP, as I was myself, I think maybe it was my ALWAYS QUESTION AUTHORITY attitude. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 11, 2025, 11:25am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/75 "2025-06-11T11:25:09Z")
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. ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [June 11, 2025, 11:46am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/76 "2025-06-11T11:46:01Z")
> [@Ynnad](#): > > I did sort of promise a response to this so my response is, “You should always agree with me.” 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. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [June 11, 2025, 12:00pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/77 "2025-06-11T12:00:39Z")
> [@Ynnad](#): > > Is the equation (5 x infinity) - (5 x infinity) = 5 a mathematical statement? If so, is it true or is it false? 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”. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 11, 2025, 12:01pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/78 "2025-06-11T12:01:38Z")
> [@The\_Other\_Waldo\_Pepper](#): > > But if you say of a given proposition ‘this is only true if zero equals five,’ Did I say that? ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [June 11, 2025, 12:08pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/79 "2025-06-11T12:08:07Z")
You said: “If we are asked to solve the equation for x we end up with the equation 0 = 5” ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 11, 2025, 12:20pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/80 "2025-06-11T12:20:58Z")
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. > [@Ynnad](#): > > 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? 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. [Previous page](https://boards.straightdope.com/t/equations-with-no-solution/1019330.md?page=3) [Next page](https://boards.straightdope.com/t/equations-with-no-solution/1019330.md?page=5)