Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$ Solving for $x,y\in\mathbb{N}$:
$$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$
This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very complicated fraction on the left hand side, and I don't even know if what I got is correct.
Is there a more mathematical way to approach this problem?

To see uniqueness,
we look at Danny's solution.
Let's see,
\begin{align}
\frac{478}{221}&=2+\frac{36}{221} &\text{2 is the only integer which will make the leftover fraction less than 1}\\
&=2+\dfrac1{\frac{221}{36}}\\&
=2+\frac1{6+\frac5{36}} &\text{Here, 6 is the only number which will make the left over fraction positive and less than 1}\\
&=2+\dfrac1{6+\dfrac1{\frac{36}{5}}}\\
&=2+\dfrac1{6+\dfrac1{7+\frac15}} &\text{and here, 7 is the only number}
\end{align}
Thus, $x=6,y=7$.
