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How to explain precisely the distinction between "using the letter $x$ as an unknown" and "using the letter $x$ as a variable"?

Is it a syntactic difference? a semantic one?

is the difference pragmatic in nature ( relative to the intentions of the person that uses the symbol : relative to "I want to find the value of $x$")?

Can I explain it in the following way :

  • $x$ is an unknown iff $x$ appears in a conditional equation

  • $x$ is a variable otherwise ( identity, defining formula of a function, etc?)

In a book ( Mathématiques de A à Z, Georges Alain , 1999) I read : " A variable is a number to which one can attribute any value one wants. An unknown is a number the possible values of which we are looking for. The oppositite of " variable" is " constant" , the opposite of " unknown" is " given").

In case this distinction would be outdated or out of use, what was the traditional explanation of this distinction?

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    $\begingroup$ It depends on the context. I would call $x$ a variable in both cases and eventually "unknown" if it is apparant that we aim to know which values it can take. Btw, there is also a distinction between free and bound variables. In the expression $\int_0^x f(u)du$ variable $x$ is free and variable $u$ is bound. $\endgroup$ – drhab Jun 1 at 14:39
  • $\begingroup$ "unknown" sounds obsolete and unclear. The line equation in the plane $y=x$, here $x,y$ are variables, but the line equation $x=1$, is here $x$ a variable or an unknown? Personally, I use "variable" in all cases. $\endgroup$ – A.Γ. Jun 1 at 14:58
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$x$, if such a notation may be introduced, is always a variable, but there are two different questions:

(1) while $x$ varies, how is another value changing?

(2) while $x$ varies, when (at which value of $x$) does something specific event happen? (A specific event: For example two values get equal.)

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People are free to use the word "variable" for unknowns too, but if we wish to distinguish the terms we can say unknowns are to be obtained, whereas variables are to be discussed generally. For example, $x$ is an unknown in $x+1=2$ but a variable in $(x^2)^\prime=2x$, or a law of physics such as $F=\frac{d}{dt}\left(m\frac{dx}{dt}\right)$. But what about $x-3+4=x+1$? Well, $x$ would be an unknown in that if you were thereby trying to solve $x-3+4=2$ by reducing it to the problem above.

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