Solve for the variable x in terms of y and z, assuming y \neq \frac{1}{2}:
xy + x = \frac{3x + 2y + z + y + 2z}{3}
We will first simplify the right side of the equation:
\begin{align*}
\frac{3x + 2y + z + y + 2z}{3} &= \frac{3x}{3} + \frac{3y}{3} + \frac{3z}{3} \\
&= x + y + z
\end{align*}
Thus, our equation becomes:
$$xy + x = x + y + z$$
We can now factor out the $x$ on the left side:
$$x(y + 1) = y + z$$
And then solve for $x$:
$$x = \boxed{\frac{y+z}{y+1}}$$