, 2024

Solution:
Adding and subtracting both equations gives
$$\begin{aligned} & \sqrt{x+2} + \sqrt{x} + \sqrt{y+2} + \sqrt{y} = 9 \\ & \sqrt{x+2} - \sqrt{x} + \sqrt{y+2} - \sqrt{y} = 1 \end{aligned}$$
Substitute $a = \sqrt{x} + \sqrt{x+2}$ and $b = \sqrt{y} + \sqrt{y+2}$. Then since $(\sqrt{x+2} + \sqrt{x})(\sqrt{x+2} - \sqrt{x}) = 2$, we have
$$\begin{gathered} a + b = 9 \\ \frac{2}{a} + \frac{2}{b} = 1 \end{gathered}$$
Dividing the first equation by the second one gives
$$ab = 18, \quad a = 3, \quad b = 6$$
Lastly, $\sqrt{x} = \frac{\sqrt{x+2} + \sqrt{x} - (\sqrt{x+2} - \sqrt{x})}{2} = \frac{3 - \frac{2}{3}}{2} = \frac{7}{6}$, so $x = \frac{49}{36}$.