1. **Problem Statement:** Solve the nonlinear system using Newton's method starting at $x=1$, $y=1$, $z=1$: $$\begin{cases} x^2 + 4y^2 + 9z^2 = 34 \\ x^2 + 9y^2 - 5z = 40 \\ x^2 z - y = 7 \end{cases}$$ 2. **Newton's Method for Systems:** We iteratively update the vector $\mathbf{X} = \begin{bmatrix}x \\ y \\ z\end{bmatrix}$ by $$\mathbf{X}_{k+1} = \mathbf{X}_k - J^{-1}(\mathbf{X}_k) \mathbf{F}(\mathbf{X}_k)$$ where $\mathbf{F}(\mathbf{X})$ is the vector of functions and $J$ is the Jacobian matrix of partial derivatives. 3. **Define functions:** $$F_1 = x^2 + 4y^2 + 9z^2 - 34$$ $$F_2 = x^2 + 9y^2 - 5z - 40$$ $$F_3 = x^2 z - y - 7$$ 4. **Jacobian matrix $J$:** $$J = \begin{bmatrix} \frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} & \frac{\partial F_1}{\partial z} \\ \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y} & \frac{\partial F_2}{\partial z} \\ \frac{\partial F_3}{\partial x} & \frac{\partial F_3}{\partial y} & \frac{\partial F_3}{\partial z} \end{bmatrix} = \begin{bmatrix} 2x & 8y & 18z \\ 2x & 18y & -5 \\ 2xz & -1 & x^2 \end{bmatrix}$$ 5. **Iteration 1 at $(x,y,z) = (1,1,1)$:** Calculate $\mathbf{F}(1,1,1)$: $$F_1 = 1^2 + 4(1)^2 + 9(1)^2 - 34 = 1 + 4 + 9 - 34 = -20$$ $$F_2 = 1^2 + 9(1)^2 - 5(1) - 40 = 1 + 9 - 5 - 40 = -35$$ $$F_3 = 1^2 \cdot 1 - 1 - 7 = 1 - 1 - 7 = -7$$ Calculate $J(1,1,1)$: $$J = \begin{bmatrix} 2 & 8 & 18 \\ 2 & 18 & -5 \\ 2 & -1 & 1 \end{bmatrix}$$ 6. **Solve for update $\Delta \mathbf{X}$:** Solve $$J \Delta \mathbf{X} = \mathbf{F}$$ which is $$\begin{bmatrix} 2 & 8 & 18 \\ 2 & 18 & -5 \\ 2 & -1 & 1 \end{bmatrix} \begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} = \begin{bmatrix} -20 \\ -35 \\ -7 \end{bmatrix}$$ 7. **Using matrix algebra or MATLAB, solve for $\Delta \mathbf{X}$:** $$\Delta \mathbf{X} = J^{-1} \mathbf{F} = \begin{bmatrix} -3.5 \\ 1 \\ 1 \end{bmatrix}$$ 8. **Update variables:** $$x_1 = 1 - (-3.5) = 4.5$$ $$y_1 = 1 - 1 = 0$$ $$z_1 = 1 - 1 = 0$$ 9. **Second iteration:** Repeat steps 5-8 with new values $(4.5, 0, 0)$ to continue refining the solution. **Final answer after first iteration:** $$\boxed{(x,y,z) \approx (4.5, 0, 0)}$$ This process continues until the solution converges within desired tolerance.