1. **State the problem:** A bookstore sells math books at 25 each and science books at 30 each. A customer buys 10 books total and pays 275 in total. We need to find how many math books and science books were bought. 2. **Set variables:** Let $x$ be the number of math books and $y$ be the number of science books. 3. **Write the system of equations:** $$\begin{cases} x + y = 10 \\ 25x + 30y = 275 \end{cases}$$ 4. **Express the system in matrix form:** $$\begin{bmatrix} 1 & 1 \\ 25 & 30 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 10 \\ 275 \end{bmatrix}$$ 5. **Solve the system using matrix methods:** Calculate the inverse of the coefficient matrix: $$A = \begin{bmatrix} 1 & 1 \\ 25 & 30 \end{bmatrix}$$ $$\det(A) = 1 \times 30 - 1 \times 25 = 5$$ $$A^{-1} = \frac{1}{5} \begin{bmatrix} 30 & -1 \\ -25 & 1 \end{bmatrix} = \begin{bmatrix} 6 & -0.2 \\ -5 & 0.2 \end{bmatrix}$$ 6. **Multiply inverse matrix by constants vector:** $$\begin{bmatrix} x \\ y \end{bmatrix} = A^{-1} \begin{bmatrix} 10 \\ 275 \end{bmatrix} = \begin{bmatrix} 6 & -0.2 \\ -5 & 0.2 \end{bmatrix} \begin{bmatrix} 10 \\ 275 \end{bmatrix}$$ Calculate each component: $$x = 6 \times 10 + (-0.2) \times 275 = 60 - 55 = 5$$ $$y = -5 \times 10 + 0.2 \times 275 = -50 + 55 = 5$$ 7. **Answer:** The customer bought 5 math books and 5 science books.