226. $\left\{\begin{array}{l}x+y=10 \\ \frac{x}{y}+\frac{y}{x}=\sqrt{5}\end{array}\right.$
226. Leonardo's solution. Excluding y, we get:

$$
x^{2}+\sqrt{50000}-200=10 x
$$

from which $x=5-\sqrt{225-\sqrt{50000}}$.

Further simplification is possible:

$$
x=15-\sqrt{125}, \quad y=\sqrt{125}-5
$$

Leonardo provides another method. Let $\frac{y}{x}=z$. Then $\frac{x}{y}=$ $=\sqrt{5}-z$. Since $\frac{y}{x} \cdot \frac{x}{y}=1$, then $z(\sqrt{5}-z)=1$ and $z=$ $=\sqrt{\frac{5}{4}}+\frac{1}{2}$. Further, $y=10-x$ and $z=\frac{10-x}{x}$, from which $\frac{10-x}{x}=$ $=\sqrt{\frac{5}{4}+\frac{1}{2}}$ gives $x=15-\sqrt{125}$.