1. **State the problem:** Solve the system of equations: $$\begin{cases} y = 2x + 1 \\ x + z = 3 \\ x + y + z = 8 \end{cases}$$ 2. **Use substitution:** From the first equation, we have $y = 2x + 1$. 3. **Substitute $y$ into the third equation:** $$x + y + z = 8 \implies x + (2x + 1) + z = 8$$ Simplify: $$3x + z + 1 = 8$$ 4. **Rewrite the second equation:** $$x + z = 3$$ 5. **Express $z$ from the second equation:** $$z = 3 - x$$ 6. **Substitute $z$ into the simplified third equation:** $$3x + (3 - x) + 1 = 8$$ Simplify: $$3x + 3 - x + 1 = 8$$ $$2x + 4 = 8$$ 7. **Solve for $x$:** $$2x + 4 = 8$$ $$2x = 8 - 4$$ $$2x = 4$$ $$x = \frac{4}{2}$$ $$x = 2$$ 8. **Find $y$ using $y = 2x + 1$:** $$y = 2(2) + 1 = 4 + 1 = 5$$ 9. **Find $z$ using $z = 3 - x$:** $$z = 3 - 2 = 1$$ **Final answer:** $$x = 2, \quad y = 5, \quad z = 1$$