1. **State the problem:** We need to find the quantities of four drugs (A, B, C, D) to distribute to four clinics so that their treatment capacities are met. 2. **Define variables:** Let $x_A, x_B, x_C, x_D$ be the quantities of drugs A, B, C, and D respectively. 3. **Write the system of equations from the table:** - Mahalapye: $2x_A + 1x_B + 3x_C + 1x_D = 25$ - Francistown: $1x_A + 3x_B + 2x_C + 2x_D = 26$ - Maun: $3x_A + 2x_B + 1x_C + 1x_D = 24$ - Gantsi: $1x_A + 2x_B + 1x_C + 3x_D = 23$ 4. **Matrix form:** $$ \begin{bmatrix} 2 & 1 & 3 & 1 \\ 1 & 3 & 2 & 2 \\ 3 & 2 & 1 & 1 \\ 1 & 2 & 1 & 3 \end{bmatrix} \begin{bmatrix}x_A \\ x_B \\ x_C \\ x_D\end{bmatrix} = \begin{bmatrix}25 \\ 26 \\ 24 \\ 23\end{bmatrix} $$ 5. **Use the inverse matrix $A^{-1}$ (given) to solve for $\mathbf{x}$:** $$ \mathbf{x} = A^{-1} \mathbf{b} $$ where $$ A^{-1} = \begin{bmatrix} \frac{1}{16} & -\frac{5}{16} & \frac{3}{8} & \frac{1}{16} \\ -\frac{5}{16} & \frac{9}{16} & \frac{1}{8} & -\frac{5}{16} \\ \frac{3}{8} & \frac{1}{8} & -\frac{1}{4} & -\frac{1}{8} \\ \frac{1}{16} & -\frac{5}{16} & -\frac{1}{8} & \frac{9}{16} \end{bmatrix} $$ 6. **Calculate each drug quantity:** - $x_A = \frac{1}{16} \times 25 - \frac{5}{16} \times 26 + \frac{3}{8} \times 24 + \frac{1}{16} \times 23 = 1.5625$ - $x_B = -\frac{5}{16} \times 25 + \frac{9}{16} \times 26 + \frac{1}{8} \times 24 - \frac{5}{16} \times 23 = 5.25$ - $x_C = \frac{3}{8} \times 25 + \frac{1}{8} \times 26 - \frac{1}{4} \times 24 - \frac{1}{8} \times 23 = 3.5$ - $x_D = \frac{1}{16} \times 25 - \frac{5}{16} \times 26 - \frac{1}{8} \times 24 + \frac{9}{16} \times 23 = 3.6875$ 7. **Interpretation:** The quantities of drugs A, B, C, and D to allocate are approximately 1.56, 5.25, 3.5, and 3.69 units respectively to meet all clinics' treatment needs.