Number of solutions of: $3x+y=5702$ 
Find the number of ordered pairs $(x,y)$ satisfying  $3x+y=5702$ in
  natural numbers restricted by: $x+y\le2003$

I don't know any method for counting number of solutions of such equations...

Hint:
Well, there's one solution for $x=1$, and one for $x=2$, and another for $x=3$ and so on.
Now, that would give you infinitely many solutions, but you have two restrictions:


*

*$x$ and $y$ are natural numbers, so they are greater than $0$.

*The sum of $x$ and $y$ must be smaller than $2003$, so that disqualifies $x=1, y=5699$, for example.


Can you continue from here to determine all solutions?
