Simultaneous Equations with Roots

28/09/2021

Solve Simultaneously

$$\sqrt{x}+y=11$$ $$x+\sqrt{y}=7$$

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Solution:

The equations imply that $x,y>0$.
Both equations have integers of the RHS
implying $x,y,\sqrt{x},\sqrt{y}$ are elements of the integers
moreover $x,y$ are perfect squares
so $0<x<7$ and $0<y<11$.
So $x\in \{1,4\}$ and $y\in \{1,4,9\}$.
Therefore $x=4$ and $y=9$
Are there any other solutions?

-oOo-