If it's acceptable practice on math.se, I'd like to really only ask this question of math educators as opposed to students or mathematical researchers. Some researchers will undoubtedly think the whole question is not worthy of consideration, and some students will not understand the issue. But if you really have something to contribute, please do.
Imagine yourself teaching a basic algebra class: maybe to grade schoolers or to high schoolers, or in my case, to adults ages 18 and up in a community college. You will encounter "problems" like the following, where for now I am intentionally leaving out words:
$$2x+3=6-x$$
The "answer" to a question like this somehow communicates that $1$ is the only solution, that $x$ needs to equal $1$, that the solution set is $\{1\}$, or $\{x\mid x=1\}$, etc.
Some of my colleagues feel that if the task was to "solve this equation", that "$x=1$" is not an acceptable final response from a student. They say that "$x=1$" is an "equivalent equation" to the original equation, because it has the same solution set. They say that to "solve this equation", to the exclusion of other ways a student might respond, is to write a set as part of an English statement. They are happy with: "The solution is $1$", "The solution set is $\{1\}$", or "The solution set is $\{x\mid x=1\}$". But they are emphatic that "$x=1$" or "The solution to the equation is that $x=1$" cannot count as appropriate responses. Answers like these don't get full credit on their exams.
This matters because we are creating a library of higher quality online problems (for WeBWorK) and we need to set a standard for how the solutions should be entered. I am opposed to having answer blanks where the student merely enters a number. So for example I am opposed to having something like "The solution is __." appear on screen and the student only fills in the number. This trains the student that those words don't matter; they don't have to write them and they won't generally pay them any attention. We could program the question to understand a whole sentence, but there are too many issues with alternatively worded correct sentences, not to mention poor spelling. Remember, these responses are to be automatically evaluated.
I support making the students have to enter "$x=1$", because it is a whole statement. And if they enter something simpler, like "$1$", it is an easy matter to shoot back an automated message that their response is not the form we are looking for. And I counter the idea that "$x=1$" has to be interpreted as an "equivalent equation" by saying that sometimes "$x=1$" is an assertion (aka an assignment) rather than an equation to solve; I'm asserting that $x$ has to equal $1$ for the equation to be true.
So my question to this community, assuming you support my position, is: can you help me make better arguments for my case? I suppose to stay within math.se guidelines, I will say that I am open to being just plain wrong, and definitive mathematical vocabulary expertise can be used to prove it.
To put it bluntly, those of your colleagues who don’t accept ‘$x=1$’ as a solution don’t speak English. That sort of hyperpedantry accomplishes nothing beyond making students think that mathematics is all about invoking the right (incomprehensible) magic formula(s). I’ll go so far as to say that I think it excessive pedantry to object to $x=\frac12\left(1\pm\sqrt5\right)$ as an answer to a question asking for the solution(s) to $x^2-x-1=0$.
It would be another matter if the question were written The solution set is __: that question clearly calls for a set. In that case, however, the script should accept $\{1\}$, $\{x\in\Bbb R:x=1\}$, $\{x:x=1\}$, $[1,1]$, and any other reasonably straightforward variant, but not $x=1$, $1$, or the like.