Solve in integers $x+\frac1{y+\frac1z}=\frac{10}7$

Question

Solve in integers $$x+\frac1{y+\frac1z}=\frac{10}7$$

My work:

1) $$\frac{xyz+x+z}{yz+1}=\frac{10}7$$

2) Solutions: $x=1, y=2, z=3$ and $x=2, y=-2, z=4$

Answer 1

When $x=1$:

$$\frac{yz+1+z}{yz+1} = \frac{10}{7}$$ $$7yz + 7 + 7z = 10yz+10$$ $$3yz-7z+3=0$$ $$z=-\frac{3}{3y-7}$$

and setting $3y-7$ equal to $-3,-1,1$, and $3$ gives the only solution $(1, 2, 3)$.

And when $x=2$: $$\frac{2yz+2+z}{yz+1} = \frac{10}{7}$$ $$14yz + 14 + 7z = 10yz+10$$ $$4yz+7z+4=0$$ $$z=-\frac{4}{4y+7}$$

and setting $4y+7$ equal to $-4, -2, -1, 1, 2$ and $4$ gives the only solution $(2, -2, 4)$.