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I often see the mix up of 'variable' and 'constant', when we describe $x$ as a variable, is it because $x$ is a symbolic element so we simply call it a 'variable' or because when solving the equation we're allowing $x$ to (in theory) change, and are simply finding the value for which the equation is true?

On the other hand could $x$ be a constant that has a particular value associated with it and we're being told that (for example) $2x+10=20$ and we can deduce that $x$ is a 'constant' and we're in a particular case where it's value is always 5? In this case when we are using a letter to signify something with a particular value, we have $x$ as essentially another name for 5? In this case the equation is always true because the value cannot change, where as in the previous case the value can change yielding a different true/false to the statement.

I have re-written this question for better clarity.

  • Related: Variable vs Constant and Variable or constant, and Dependence among variables. A site search may turn up more. – Blue May 02 '22 at 16:39
  • Basically you have the function $f(x)=2x+10$ with variable $x$. You can set this function equal to another function $g(x)=20$. Then you are looking for the value(s) of x which fulfill the constraint. So the variable $x$ become more restricted. In this case from $x\in \mathbb R$ to $x=5$. – callculus42 May 02 '22 at 16:49
  • Could you imagine having the same problem with pronouns? "He" is a variable noun. One day it stands for Dick, the next for Harry. That doesn't make "he" constant. – B. Goddard May 02 '22 at 22:02
  • Importantly $x$ is a constant because the expression is always true, the expression can be true/false for different values if $x$ varies. –  May 02 '22 at 22:41
  • @B.Goddard exploring the value of a variable when it is true/false is different from looking at $x$ constant where it's ALWAYS true. –  May 02 '22 at 22:43
  • This does depend very much on context - if the text mentioned $x$ before this point (usually within the same problem/proof/section/etc.), that first mention usually sets up what sort of thing $x$ means. – aschepler May 02 '22 at 23:03
  • The equation $\sin^2 x + \cos ^2 x =1$ is always true, and $x$ is not a constant. – B. Goddard May 03 '22 at 11:10
  • My answers are here: Variable or constant? and Types of equations. – ryang May 03 '22 at 12:35
  • @B.Goddard it's always true for every value of $x$, and perhaps there's many values for which the value is true or false but generally we say something is 'true for all values' not just 'true' or 'false' without assigning to it first, generally we can't give it a truth value without assigning to it first if $x$ is a variable, if its constant than the expression is always true or false otherwise some equations would have well defined truth values and some won't (I cannot say whether $2x+10=20$ is true or false without first knowing $x$. –  May 03 '22 at 18:43
  • We don't like to make statements about truth until we can concretely talk about the objects involved. –  May 03 '22 at 18:45

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It is more proper to call the $x$ in $2x + 10 = 20$ an unknown rather than a variable. A proper variable is something that could actually take on different values, like the $t$ in the function definition $a(t) = t^2e^t$.

So then why does $x$ often get called a variable in the above equation anyways? Well, I think we often conceptually think of the two sides of the equation as being functions of $x$. That is, "Find the value of the variable $x$ such that $f(x) = g(x)$, where $f(x) = 2x + 10$ and $g(x) = 20$". If posed like that, $x$ does represent a variable. And since humans are notoriously sloppy with terminology, it then also gets called a variable when posed as "Find the value of $x$ such that $2x + 10 = 20$".

eyeballfrog
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  • so if we know the expression is actually true, then $x$ is better called an unknown, and if we are just finding what $x$ in theory makes it true, $x$ is a variable? –  May 02 '22 at 17:04