A 3-minute algebra problem I have just taken advance math test level 2 and there are several problems that have been bugging me. This is the first question:

If $x,y>0$, then determine the value of $x$ that satisfies the system
  of equations: \begin{align} x^2+y^2-xy&=3\\ x^2-y^2+\sqrt{6}y&=3\\
\end{align}

I can answer this problem using a 'standard' algebra but it takes time more than 3 minutes. For the sake of curiosity, is there a way to answer this problem less than 3 minutes?

Take G.Bach's hint so that $2y=x+\sqrt 6$
Now add the two equations to obtain $$2x^2+(\sqrt 6-x)y=6$$ Substitute for $y$: $$2x^2+\frac 12(\sqrt 6-x)(\sqrt 6+x)=6=\frac 32 x^2+3$$ 
So that $x^2=2$
