1. **State the problem:** Solve the simultaneous equations using the matrix method: $$2n + 3m = 12$$ $$4m - 2n = 5$$ 2. **Write the system in matrix form:** $$\begin{bmatrix} 2 & 3 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} n \\ m \end{bmatrix} = \begin{bmatrix} 12 \\ 5 \end{bmatrix}$$ 3. **Find the determinant of the coefficient matrix:** $$D = (2)(4) - (-2)(3) = 8 + 6 = 14$$ Since $D \neq 0$, the system has a unique solution. 4. **Find the inverse of the coefficient matrix:** $$A^{-1} = \frac{1}{D} \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix} = \frac{1}{14} \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix}$$ 5. **Multiply the inverse matrix by the constants vector:** $$\begin{bmatrix} n \\ m \end{bmatrix} = A^{-1} \begin{bmatrix} 12 \\ 5 \end{bmatrix} = \frac{1}{14} \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 12 \\ 5 \end{bmatrix}$$ Calculate the multiplication: $$\begin{bmatrix} (4)(12) + (-3)(5) \\ (2)(12) + (2)(5) \end{bmatrix} = \begin{bmatrix} 48 - 15 \\ 24 + 10 \end{bmatrix} = \begin{bmatrix} 33 \\ 34 \end{bmatrix}$$ 6. **Divide by the determinant:** $$\begin{bmatrix} n \\ m \end{bmatrix} = \frac{1}{14} \begin{bmatrix} 33 \\ 34 \end{bmatrix} = \begin{bmatrix} \frac{33}{14} \\ \frac{34}{14} \end{bmatrix} = \begin{bmatrix} 2.357 \text{ (approx)} \\ 2.429 \text{ (approx)} \end{bmatrix}$$ **Final answer:** $$n \approx 2.36, \quad m \approx 2.43$$