1. **Problem Statement:** Solve the system of linear equations: $$\begin{cases} 3x + 2y + z = 2 \\ 4x + 2y + 2z = 8 \\ x - y + z = 4 \end{cases}$$ 2. **Method:** We will use the elimination or substitution method to find $x$, $y$, and $z$. 3. **Step 1: Write the system clearly:** $$3x + 2y + z = 2 \quad (1)$$ $$4x + 2y + 2z = 8 \quad (2)$$ $$x - y + z = 4 \quad (3)$$ 4. **Step 2: Eliminate $y$ by subtracting (1) from (2):** $$(4x + 2y + 2z) - (3x + 2y + z) = 8 - 2$$ Simplify: $$4x - 3x + 2y - 2y + 2z - z = 6$$ $$x + z = 6 \quad (4)$$ 5. **Step 3: Express $z$ from (4):** $$z = 6 - x \quad (5)$$ 6. **Step 4: Substitute $z$ from (5) into (3):** $$x - y + (6 - x) = 4$$ Simplify: $$x - y + 6 - x = 4$$ $$-y + 6 = 4$$ Subtract 6 from both sides: $$-y = 4 - 6$$ $$-y = -2$$ Multiply both sides by $\cancel{-1}$: $$\cancel{-1} \times -y = \cancel{-1} \times -2$$ $$y = 2 \quad (6)$$ 7. **Step 5: Substitute $y=2$ and $z=6 - x$ into (1):** $$3x + 2(2) + (6 - x) = 2$$ Simplify: $$3x + 4 + 6 - x = 2$$ $$2x + 10 = 2$$ Subtract 10 from both sides: $$2x = 2 - 10$$ $$2x = -8$$ Divide both sides by $\cancel{2}$: $$\cancel{2}x / \cancel{2} = -8 / \cancel{2}$$ $$x = -4$$ 8. **Step 6: Find $z$ using (5):** $$z = 6 - (-4) = 6 + 4 = 10$$ 9. **Final solution:** $$\boxed{x = -4, y = 2, z = 10}$$ 10. **Consistency check:** Substitute values into all equations to verify: - Equation (1): $3(-4) + 2(2) + 10 = -12 + 4 + 10 = 2$ ✓ - Equation (2): $4(-4) + 2(2) + 2(10) = -16 + 4 + 20 = 8$ ✓ - Equation (3): $-4 - 2 + 10 = 4$ ✓ The system is consistent and has a unique solution.