This is the question:
$$\frac{36}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 42-9\sqrt{x}-\sqrt{y}$$
This is from a timed competition, and I would like to know the fastest way to do it. I'm not sure, but is there a faster way than to replace $\sqrt{x}$ and $\sqrt{y}$ with other variables?
Rewrite: $$\Big(\frac{36}{\sqrt{x}} + 9\sqrt{x}\Big)+\Big(\frac{9}{\sqrt{y}}+\sqrt{y}\Big)=42$$
Now by Am-Gm inequality ($a+b\geq 2\sqrt{ab}$) we have $$\frac{36}{\sqrt{x}} + 9\sqrt{x}\geq 36$$ and $$\frac{9}{\sqrt{y}} +\sqrt{y}\geq 6$$
So we must have equality case. Thus $x =4$ and $y= 9$.
