Does other solutions exist for $29x+30y+31z = 366$? I was asked this trick question:
If $29x + 30y + 31z = 366$ then what is $x+y+z=?$ 
The answer is $12$ and it is said to be so because $29$ , $30$  and $31$ are respectively the number of days of months in a leap year. Therefore $x + y + z$ must be $12$, the total number of months.
How accurate is this? Is it possible to say so with just a single equation? Are there not other solutions to the equation? If yes, how can one proceed to find other solutions?

Here is the solution to Alan's "middle case", where the three variables are allowed to be integers: 
Choose any integer for $z$ and demand that $x=-y$ (so that $x+y=0$); then $x+y+z=z$, so if we can find a value for $y$ that satisfies the 366 constraint, we show that the $x+y+z$ can be any integer. 
This is possible: observe that
$$29x+30y+31z ~=~ 29(x+y)+y+31z ~=~ y+31z,$$
so that if we choose $y=366-31z$, we get $(366-31z)+31z=366.$
