1. **State the problem:** A student buys notebooks and pens. Each notebook costs $4 and each pen costs $1. The student buys 18 items in total and spends 54 in total cost. We need to find how many notebooks and pens the student bought. 2. **Set variables:** Let $x$ be the number of notebooks and $y$ be the number of pens. 3. **Write the system of equations:** $$\begin{cases} x + y = 18 \\ 4x + y = 54 \end{cases}$$ 4. **Express the system in matrix form:** $$\begin{bmatrix} 1 & 1 \\ 4 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 18 \\ 54 \end{bmatrix}$$ 5. **Find the inverse of the coefficient matrix:** The determinant is $\det = 1 \times 1 - 4 \times 1 = 1 - 4 = -3$. The inverse matrix is: $$\frac{1}{-3} \begin{bmatrix} 1 & -1 \\ -4 & 1 \end{bmatrix} = \begin{bmatrix} -\frac{1}{3} & \frac{1}{3} \\ \frac{4}{3} & -\frac{1}{3} \end{bmatrix}$$ 6. **Multiply the inverse matrix by the constants vector:** $$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -\frac{1}{3} & \frac{1}{3} \\ \frac{4}{3} & -\frac{1}{3} \end{bmatrix} \begin{bmatrix} 18 \\ 54 \end{bmatrix} = \begin{bmatrix} -\frac{1}{3} \times 18 + \frac{1}{3} \times 54 \\ \frac{4}{3} \times 18 - \frac{1}{3} \times 54 \end{bmatrix}$$ Calculate each: $$x = -6 + 18 = 12$$ $$y = 24 - 18 = 6$$ 7. **Answer:** The student bought 12 notebooks and 6 pens.