Is there any simple analytic method for solving $\sqrt{x}+y=7$ and $x+\sqrt{y}=11$ simultaneously.

Question

I am thinking of a nice and simple analytic method to solve the following equations simultaneously:

$$\sqrt x+y=7;\\x+\sqrt y=11.$$

To my suprise I can't. But, I solve the system numerically using Newton's method and the solution is $4$ and $9$.

Is there any simple way of solving the above system analytically?

Answer 1

Hint: Raise to power of two both equation and eliminate the squares.