There exists a minimum value such that for every positive integer n, there exists x, y such that n = 4x + 7y.
There exists a minimum number that for every real number, we want to prove it can be expressed as 4x + 7y, where x and y are some natural numbers.
One of the first things to learn about proofs is that the exact wording of a theorem can make a huge difference. That's why it's essential that you quote the exact assignment. It may also be valuable if you tell us a little about the context -- assuming this is for a course that introduces the idea of proofs, what methods, and what axioms or theorems, have you learned that you can use? And are there any specific instructions or hints provided?
There are some huge differences between your two versions, as marked above, but they don't touch the issue we are all concerned about, which is the meaning of "minimum number".
Now, my guess is that it's supposed to be something like this:
There exists a minimum number m such that, for every positive integer n >= m, there exist natural numbers x and y such that n = 4x + 7y.
At several points you may have to correct me. (Note that positive integer and natural number mean the same thing, but real numbers are very different.)
I would start by doing just what we are trying to do: clarify the meaning of the problem, before attacking it.
One important way will be to "play" with the concept to get a feel for what it is about. You might start by picking a small number n and convincing yourself either that it can't be written as 4x + 7y (presumably because it is less than the minimum mentioned), or that it can (by actually doing it). This will give you a good idea of what might go wrong (in the first case), or why it is true (in the second case). You can use those ideas in the proof -- maybe!
Once you've done that, it's quite possible that you will need to actually find the minimum. That, too, you may just do by experiment.
But, again, I can't talk about any details until I'm sure what the problem really is, which means you must show us the exact wording (yes, even if that is in another language).
