Say I have a question that says:
"answer the question in terms of x" where x is a variable.
vs.
"answer the questions in terms of n" where n is any constant.
What is the difference between the two? ( in terms of definition )
Edit: To try and clarify, I was wondering if there was any discernible difference in the problem solving process of the two situations ( whether it be theoretical or bound by mathematical definition) . However, the answer to the question has been given below ( referring to stonebrakermatt's and Biderman's answers ).
Notice that a variable can represent a constant, i.e. $5x+10 = 25$. $x = 3$, so $x$ clearly is constant; however, $x$ is a variable.
When you state that $x$ is a variable, I will assume you mean something along the lines of $y = f(x)$, where $y$ can assume a range of values.
As Joshua said, there is no difference when it comes to problem solving.
Consider the following, where we solve for $45$ and $y$ in terms of $x$:
$5x+10 = 25 \Rightarrow 15x = 45$
$xy + 10 = 3 \Rightarrow y = \frac{-7}{x}$
In the first example, $x$ will not vary, while it will do so in the second example.
While it will not vary in way of problem-solving, the results that you yield from such a calculation are distinct. Any expression involving constants only (i.e. $\sqrt{2^{\log_34}}$) will always be a constant, no matter how complex, even if that constant is represented as a variable. This is not the case when you solve for something in terms of a nonconstant variable.
EDIT: You can often solve many types of equations in terms of an arbitrary constant, which seems to be what your example is getting at. For example, consider $x^2 \equiv 1$ (mod $10$) can give you the solutions
$x = 10n + 1$ and $x = 10n + 9$, where $n$ is any integer.