Fombitz thought you meant all solutions, but you probably meant 
only solutions (x,y) where x and y are both whole numbers:

3x + 7y = 999

Write all the numbers in terms of the nearest multiple to the absolute
value of the coefficient of the variable with the smallest absolute 
value that are not already a multiple of it:

Of the numbers 3, 7, 333, only  7 is not a multiple of 3, so we write 
7 in terms of its nearest multiple of 3, which is 6. So we write
7 as 6+1
 

3x+(6+1)y = 999
  3x+6y+y = 999

WE divide every term by 3

  x+2y+y/3 = 333

y/3 = 333-x

Since 333-x is a non-negative integer, so is y/3.

Let the integer be A, then

333-x = A  and y/3 = A

So y = 3A

We substitute that in


3x + 7y = 999

3x + 7(3A) = 999

3x + 21A = 999

Divide through by 3

x + 7A = 333

x = 333-7A

Now since x >= 0,  

333-7A >= 0

-7A >= -333
  A <= 333/7 = 47 4/7
  A <= 47

Also since y >= 0

  3A >= 0 
   A >= 0

So there are 48 solutions from A = 0 through 47 inclusively.

Edwin