The problem is as follows:
Let $x$ and $y$ integers which satisfy the following equations: $$x+y-\sqrt{xy}=7$$ $$x^2+y^2+xy=133$$ Find the value of $\;|x-y|.$
I'm stuck on this problem due the fact that there appears a square root of $xy$ and the squares of both $x$ and $y$, hence the system cannot be solved using the regular methods. Moreover I don't know how to approach the absolute value.
The answer which would help me the most is one which addresses some theoretical basis about absolute value and steps which would led me to find $x$ and $y$.
some theoretical basis
One clue (which I used in my answer) is that both the equations and the required value are symmetric in $x,y$. When that's the case, often times it helps to work in terms of the elementary symmetric polynomials, instead, in this case $x+y$ and $xy$. $\endgroup$ – dxiv Dec 18 '17 at 21:01