Not Continued Fractions

'Not Continued Fractions' printed from https://nrich.maths.org/

Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$
Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.