<s>[INST] Finding the integer solutions of the equation $3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14$ $
3\sqrt {x + y}  + 2\sqrt {8 - x}  + \sqrt {6 - y}  = 14
$ . 
I already solved this using the  Cauchy–Schwarz inequality and got $x=4$ and $y=5$. But I'm sure there is a prettier, simpler solution to this and I was wondering if anyone could suggest one.
 [/INST] This is a trivial solution so that we get answer in integer
8 - x and 6 - y are under the root so x <8 and y <6 , for making 6 - y to be perfect square y=2 or 5. and similarly x=4 or 7.
Now x+y is also under the root so x+y has to be perfect square. So we get two pairs of (x, y)=(7,2) and (2,5) but (7,2) will not satisfy the equation so ur answer is x=4, y=5.
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