1. **State the problem:** Solve the system of linear equations using the Jacobi Method: $$\begin{cases} 5x - y + z = 10 \\ 2x + 8y - z = 12 \\ x - y + 4z = 6 \end{cases}$$ 2. **Jacobi Method formula:** Rewrite each equation to isolate each variable: $$x = \frac{10 + y - z}{5}$$ $$y = \frac{12 - 2x + z}{8}$$ $$z = \frac{6 - x + y}{4}$$ 3. **Initial guess:** Start with $x^{(0)}=0$, $y^{(0)}=0$, $z^{(0)}=0$. 4. **Iteration 1:** $$x^{(1)} = \frac{10 + 0 - 0}{5} = 2$$ $$y^{(1)} = \frac{12 - 2(0) + 0}{8} = 1.5$$ $$z^{(1)} = \frac{6 - 0 + 0}{4} = 1.5$$ 5. **Iteration 2:** $$x^{(2)} = \frac{10 + 1.5 - 1.5}{5} = \frac{10}{5} = 2$$ $$y^{(2)} = \frac{12 - 2(2) + 1.5}{8} = \frac{12 - 4 + 1.5}{8} = \frac{9.5}{8} = 1.1875$$ $$z^{(2)} = \frac{6 - 2 + 1.5}{4} = \frac{5.5}{4} = 1.375$$ 6. **Iteration 3:** $$x^{(3)} = \frac{10 + 1.1875 - 1.375}{5} = \frac{9.8125}{5} = 1.9625$$ $$y^{(3)} = \frac{12 - 2(2) + 1.375}{8} = \frac{12 - 4 + 1.375}{8} = \frac{9.375}{8} = 1.171875$$ $$z^{(3)} = \frac{6 - 1.9625 + 1.171875}{4} = \frac{5.209375}{4} = 1.302344$$ 7. **Iteration 4:** $$x^{(4)} = \frac{10 + 1.171875 - 1.302344}{5} = \frac{9.869531}{5} = 1.973906$$ $$y^{(4)} = \frac{12 - 2(1.9625) + 1.302344}{8} = \frac{12 - 3.925 + 1.302344}{8} = \frac{9.377344}{8} = 1.172168$$ $$z^{(4)} = \frac{6 - 1.973906 + 1.172168}{4} = \frac{5.198262}{4} = 1.299566$$ 8. **Check convergence:** Values are converging. Continue until values stabilize to 3 decimal places. 9. **Final approximate solution:** $$x \approx 1.974, \quad y \approx 1.172, \quad z \approx 1.300$$ These values satisfy the system to 3 decimal places using the Jacobi Method.