Mathematical teaser equation

Hello everyone!

I hope everyone has had a relaxing summer.

Anyway, I have ventured again into a realm of math that is far beyond my abilities, in other words I am trying to solve some mathematical problems from the book *Challenging Mathematical Teasers* by J.A.H. Hunter

The word problem I am currently solving leads to the following two equations

$\displaystyle

x^2-y^2=z^3

$

and

$\displaystyle \frac{x}{y}=\frac{y}{z} $

by substiuting we will have $\displaystyle x^2-xz=z^3 $

This is about as far as I got, but then the solutions says something that I can't quite understand.

*This equation is fully satisfied by: $\displaystyle x-z=mz$, and x=$\displaystyle \frac{z^2}{m}$, where m is any rational number. *

Could someone pleade exmplain the above thing to me? Does the author of this book use some systematic approach to solving these kind of equations with 3 unknowns or is it just the mathematician's creativity?

All responses are appreciated.