1. **State the problem:** Solve the system of equations: $$3x - 2y + 4z = 5$$ $$\frac{1}{2}x + 4y = 4$$ $$4y - 2z = 21$$ 2. **Rewrite the system for clarity:** $$3x - 2y + 4z = 5 \quad (1)$$ $$\frac{1}{2}x + 4y = 4 \quad (2)$$ $$4y - 2z = 21 \quad (3)$$ 3. **Express $x$ from equation (2):** $$\frac{1}{2}x + 4y = 4$$ Subtract $4y$ from both sides: $$\frac{1}{2}x = 4 - 4y$$ Multiply both sides by 2: $$x = 2(4 - 4y) = 8 - 8y$$ 4. **Express $z$ from equation (3):** $$4y - 2z = 21$$ Subtract $4y$ from both sides: $$-2z = 21 - 4y$$ Divide both sides by $-2$: $$z = \frac{\cancel{-2}z}{\cancel{-2}} = \frac{4y - 21}{2}$$ 5. **Substitute $x$ and $z$ into equation (1):** $$3x - 2y + 4z = 5$$ Substitute $x = 8 - 8y$ and $z = \frac{4y - 21}{2}$: $$3(8 - 8y) - 2y + 4 \left( \frac{4y - 21}{2} \right) = 5$$ 6. **Simplify the equation:** $$24 - 24y - 2y + 2(4y - 21) = 5$$ $$24 - 24y - 2y + 8y - 42 = 5$$ Combine like terms: $$24 - 24y - 2y + 8y - 42 = 5$$ $$24 - 18y - 42 = 5$$ $$-18y - 18 = 5$$ 7. **Solve for $y$:** Add 18 to both sides: $$-18y = 23$$ Divide both sides by $-18$: $$y = \frac{\cancel{-18}y}{\cancel{-18}} = -\frac{23}{18}$$ 8. **Find $x$ using $y$:** $$x = 8 - 8y = 8 - 8 \left(-\frac{23}{18}\right) = 8 + \frac{184}{18} = 8 + \frac{92}{9} = \frac{72}{9} + \frac{92}{9} = \frac{164}{9}$$ 9. **Find $z$ using $y$:** $$z = \frac{4y - 21}{2} = \frac{4 \left(-\frac{23}{18}\right) - 21}{2} = \frac{-\frac{92}{18} - 21}{2} = \frac{-\frac{92}{18} - \frac{378}{18}}{2} = \frac{-\frac{470}{18}}{2} = -\frac{470}{36} = -\frac{235}{18}$$ **Final solution:** $$x = \frac{164}{9}, \quad y = -\frac{23}{18}, \quad z = -\frac{235}{18}$$