Subtleties of "unknown" vs. "variable" I'm trying to pin down the difference between "unknown" and "variable". I have always understood that in the equations $2x+1=10$ or $x^2+5x+6=0$, $x$ is an unknown (short for "unknown constant"), since its value can be determined. In the expression $2x+1$, however, $x$ can take any value, therefore it is a variable.

What about in the equation $2x+3y=10$? $x$ and $y$ can both take infinitely many values, but once one is fixed, the other becomes fixed. Does this mean they are both variables? Does it mean that one (say $x$) is a variable, but the other is an unknown (since its value is determined by the variable)?

I'd appreciate some insight. Thanks.
It depends what your question is. While they are kind of interchangeable, these two terms are used in different contexts.  Unknown is usually employed in equations,so for example you could ask how to solve the equation $2x=1$,where $x$ is the unknown. On the other hand the term variable is more used in case of functions. You could ask for example what is the second derivative according to the $x$ variable of the function $f(x,y)=x^2+y+2$. Hope that clears things up a bit.
I would say...a variable is an unknown but an unknown doesn't necessarily have to be a variable. A variable means it could be any number, it is not fixed but a unknown means it is a specific number that we do not know as yet. Therefore a variable is an unknown because it could be any number but an unknown doesn't have to be a variable because it is a fixed number that we do not know. Hope that makes sense.