Represent a line as an equality with 0

I have the following line in $\mathbb{R}^3$:

$$r\equiv x=y=z$$

I can also represent $r$ with a set of equalities like:

$$r\equiv\begin{cases}2x = y + z \\2y = x + z \\2z = x + y\end{cases}$$

However, I need to express it in a form like $f(x,y,z)=0$. I tried with conditions like $2x-y-z=0$, but it is necessary to characterize the line with all of the above equations. Thus, is there a way to write $r$ as an equality of a function of $x,y,z$ to 0?