1. **State the problem:** Solve the equation $$\frac{1}{3}(x-3) - x + \frac{1}{3} + 3 + \frac{x}{3} = \frac{1}{3} - z - \frac{x}{3} + \frac{x}{3} + 1$$ for $x$ and $z$. 2. **Simplify both sides:** Left side: $$\frac{1}{3}(x-3) - x + \frac{1}{3} + 3 + \frac{x}{3} = \frac{x}{3} - 1 - x + \frac{1}{3} + 3 + \frac{x}{3}$$ Combine like terms: $$\frac{x}{3} + \frac{x}{3} - x + (-1 + \frac{1}{3} + 3) = \frac{2x}{3} - x + \frac{7}{3}$$ Rewrite $-x$ as $-\frac{3x}{3}$: $$\frac{2x}{3} - \frac{3x}{3} + \frac{7}{3} = -\frac{x}{3} + \frac{7}{3}$$ Right side: $$\frac{1}{3} - z - \frac{x}{3} + \frac{x}{3} + 1 = \frac{1}{3} - z + 1$$ Simplify constants: $$\frac{1}{3} + 1 = \frac{4}{3}$$ So right side is: $$\frac{4}{3} - z$$ 3. **Set simplified sides equal:** $$-\frac{x}{3} + \frac{7}{3} = \frac{4}{3} - z$$ 4. **Isolate $z$:** $$z = \frac{4}{3} + \frac{x}{3} - \frac{7}{3} = \frac{4}{3} - \frac{7}{3} + \frac{x}{3} = -1 + \frac{x}{3}$$ 5. **Final answer:** $$\boxed{z = -1 + \frac{x}{3}}$$ This expresses $z$ in terms of $x$. Since the equation has two variables and one equation, this is the simplest form relating $z$ and $x$.