What is the difference between the domain of a variable and the domain for an equation? domain of a variable: The given set of numbers that the variable may represent.
What does it mean for an equation to have a domain? 
Suppose the domain for an equation is {1, 2, 3, 4, ...}
How is that different from the variable in the equation having the same domain?
For example does it make sense to talk about finding the solution set of an equation over a given domain?
Say for the equation: $x + x = 2x$ over the domain {1, 2, 3, 4, ...}.
 A: I wouldn't say that an equation ever has a domain. It does, however, make sense to talk about finding the solution set of an equation over a given domain.
For instance, the solution set of $$x + x = 2x$$ over the domain $$\{1,2,3,\ldots\}$$ would be precisely the above set, because these are precisely those values of $x$ that make the statement true.
More generally, it makes sense to speak of the solution set of any statement in which $x$ is a free variable (so long as the range over which $x$ varies is clear). For example, the statement "$x>0$ and $x<3$" has solution set equal to the open interval $(0,3),$ as long as its clear that $x$ varies over $\mathbb{R}.$
However, I think this is a moderately sloppy way of talking. What is the solution set of $$x+y=0?$$
Is it all the values $x$ such that $x+y=0$? If so, then $\{-y\}$ is the solution set.
Or is it the set of ordered pairs $(x,y)$ such that $x+y=0$? If so, then $\{(x,-x) \mid x \in \mathbb{R}\}$ is the solution set.
Therefore, I think it is clearer to say: "Consider the set of all real $x$ that satisfy the equation," as compared to, "Consider the solution set of the equation over the real numbers."
A: It doesn't make any sense to say an equation has a domain.  An equation may be comprised of many variables each of which has its own domain.  A function also has a domain consisting of whatever values it may be applied to. 
