1. **State the problem:** Solve the system of linear equations using the Gauss-Seidel iteration method: $$\begin{cases} 2x + y + z = 4 \\ x + 2y + z = 4 \\ x + y + 2z = 4 \end{cases}$$ 2. **Rewrite each equation to isolate each variable:** $$x = \frac{4 - y - z}{2}$$ $$y = \frac{4 - x - z}{2}$$ $$z = \frac{4 - x - y}{2}$$ 3. **Choose initial guesses:** Let $x^{(0)}=0$, $y^{(0)}=0$, $z^{(0)}=0$. 4. **Iterate using Gauss-Seidel:** - Iteration 1: $$x^{(1)} = \frac{4 - 0 - 0}{2} = 2$$ $$y^{(1)} = \frac{4 - 2 - 0}{2} = 1$$ $$z^{(1)} = \frac{4 - 2 - 1}{2} = 0.5$$ - Iteration 2: $$x^{(2)} = \frac{4 - 1 - 0.5}{2} = 1.25$$ $$y^{(2)} = \frac{4 - 1.25 - 0.5}{2} = 1.125$$ $$z^{(2)} = \frac{4 - 1.25 - 1.125}{2} = 0.8125$$ - Iteration 3: $$x^{(3)} = \frac{4 - 1.125 - 0.8125}{2} = 1.03125$$ $$y^{(3)} = \frac{4 - 1.03125 - 0.8125}{2} = 1.078125$$ $$z^{(3)} = \frac{4 - 1.03125 - 1.078125}{2} = 0.9453125$$ 5. **Continue iterations until values converge:** After several iterations, the solution converges to approximately: $$x \approx 1, \quad y \approx 1, \quad z \approx 1$$ 6. **Final answer:** $$\boxed{(x, y, z) = (1, 1, 1)}$$