1. **State the problem:** We want to find the number of loaves of White, Wholemeal, and Multigrain bread produced given the total amounts of flour, yeast, and water used. 2. **Set up the system of equations:** Let $x$, $y$, and $z$ be the number of loaves of White, Wholemeal, and Multigrain bread respectively. From the table: - Flour: $2x + 1y + 1z = 160$ - Yeast: $1x + 2y + 3z = 200$ - Water: $1x + 2y + 4z = 260$ 3. **Matrix form:** $$\begin{bmatrix}2 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & 4\end{bmatrix} \begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}160 \\ 200 \\ 260\end{bmatrix}$$ 4. **Find the inverse of the coefficient matrix:** Let $$A = \begin{bmatrix}2 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & 4\end{bmatrix}$$ Calculate determinant $|A|$: $$|A| = 2(2 \times 4 - 3 \times 2) - 1(1 \times 4 - 3 \times 1) + 1(1 \times 2 - 2 \times 1)$$ $$= 2(8 - 6) - 1(4 - 3) + 1(2 - 2) = 2(2) - 1(1) + 1(0) = 4 - 1 + 0 = 3$$ 5. **Calculate the adjugate matrix $\text{adj}(A)$:** Calculate cofactors: - $C_{11} = +(2 \times 4 - 3 \times 2) = 8 - 6 = 2$ - $C_{12} = -(1 \times 4 - 3 \times 1) = -(4 - 3) = -1$ - $C_{13} = +(1 \times 2 - 2 \times 1) = 2 - 2 = 0$ - $C_{21} = -(1 \times 4 - 3 \times 1) = -(4 - 3) = -1$ - $C_{22} = +(2 \times 4 - 1 \times 1) = 8 - 1 = 7$ - $C_{23} = -(2 \times 2 - 1 \times 1) = -(4 - 1) = -3$ - $C_{31} = +(1 \times 2 - 2 \times 1) = 2 - 2 = 0$ - $C_{32} = -(2 \times 2 - 1 \times 1) = -(4 - 1) = -3$ - $C_{33} = +(2 \times 2 - 1 \times 1) = 4 - 1 = 3$ So, $$\text{adj}(A) = \begin{bmatrix}2 & -1 & 0 \\ -1 & 7 & -3 \\ 0 & -3 & 3\end{bmatrix}^T = \begin{bmatrix}2 & -1 & 0 \\ -1 & 7 & -3 \\ 0 & -3 & 3\end{bmatrix}$$ 6. **Find the inverse:** $$A^{-1} = \frac{1}{|A|} \text{adj}(A) = \frac{1}{3} \begin{bmatrix}2 & -1 & 0 \\ -1 & 7 & -3 \\ 0 & -3 & 3\end{bmatrix}$$ 7. **Calculate the solution vector:** $$\begin{bmatrix}x \\ y \\ z\end{bmatrix} = A^{-1} \begin{bmatrix}160 \\ 200 \\ 260\end{bmatrix} = \frac{1}{3} \begin{bmatrix}2 & -1 & 0 \\ -1 & 7 & -3 \\ 0 & -3 & 3\end{bmatrix} \begin{bmatrix}160 \\ 200 \\ 260\end{bmatrix}$$ Calculate each component: - $x = \frac{1}{3}(2 \times 160 - 1 \times 200 + 0 \times 260) = \frac{1}{3}(320 - 200 + 0) = \frac{120}{3} = 40$ - $y = \frac{1}{3}(-1 \times 160 + 7 \times 200 - 3 \times 260) = \frac{1}{3}(-160 + 1400 - 780) = \frac{460}{3} \approx 153.33$ - $z = \frac{1}{3}(0 \times 160 - 3 \times 200 + 3 \times 260) = \frac{1}{3}(0 - 600 + 780) = \frac{180}{3} = 60$ 8. **Interpretation:** - White bread loaves: $40$ - Wholemeal bread loaves: approximately $153$ - Multigrain bread loaves: $60$ **Final answer:** $$\boxed{\text{White} = 40, \quad \text{Wholemeal} = 153, \quad \text{Multigrain} = 60}$$