1. **State the problem:** Solve the simultaneous equations for integers $x, y, z$: $$\begin{cases} x - 2y = -3 \\ 2y + z = 3 \\ z - 2x = -3 \end{cases}$$ 2. **Use substitution or elimination:** From equation (1), express $x$ in terms of $y$: $$x = -3 + 2y$$ 3. **Substitute $x$ into equation (3):** $$z - 2(-3 + 2y) = -3$$ $$z + 6 - 4y = -3$$ $$z = -3 - 6 + 4y$$ $$z = -9 + 4y$$ 4. **Substitute $z$ into equation (2):** $$2y + (-9 + 4y) = 3$$ $$2y - 9 + 4y = 3$$ $$6y - 9 = 3$$ $$6y = 3 + 9$$ $$6y = 12$$ 5. **Solve for $y$:** $$y = \frac{12}{6}$$ $$y = 2$$ 6. **Find $x$ using $y=2$:** $$x = -3 + 2(2)$$ $$x = -3 + 4$$ $$x = 1$$ 7. **Find $z$ using $y=2$:** $$z = -9 + 4(2)$$ $$z = -9 + 8$$ $$z = -1$$ **Final answer:** $$\boxed{(x, y, z) = (1, 2, -1)}$$