I am working on the following question:

For all $x \in \mathbb{R}$, $x \neq 6$, there exists a unique real number $y$ such that $xy+x=6y$.

Now I have the existence part. That there exists a $$y=\frac{x}{6-x}.$$
To show uniqueness I know that I must show that if there is any other number say $z=x/(6-x)$, then $z$ must equal $y$. But I am not exactly sure how to show that part.