Paper Prices 264D54

1. **State the problem:** We are given two equations representing the prices of reams of A4 and A3 paper: $$3x + 2y = 42$$ $$2x + 5y = 61$$ where $x$ is the price of one ream of A4 paper and $y$ is the price of one ream of A3 paper. 2. **Formula and rules:** To solve this system of linear equations, we use the matrix inverse method: $$\begin{pmatrix}x \\ y\end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix} \begin{pmatrix}e \\ f\end{pmatrix}$$ where the system is: $$a x + b y = e$$ $$c x + d y = f$$ 3. **Identify values:** From the equations: $$a=3, b=2, c=2, d=5, e=42, f=61$$ 4. **Calculate determinant:** $$ad - bc = 3 \times 5 - 2 \times 2 = 15 - 4 = 11$$ 5. **Calculate inverse matrix multiplied by constants:** $$\begin{pmatrix}d & -b \\ -c & a\end{pmatrix} = \begin{pmatrix}5 & -2 \\ -2 & 3\end{pmatrix}$$ 6. **Multiply inverse matrix by constants:** $$\begin{pmatrix}5 & -2 \\ -2 & 3\end{pmatrix} \begin{pmatrix}42 \\ 61\end{pmatrix} = \begin{pmatrix}5 \times 42 + (-2) \times 61 \\ -2 \times 42 + 3 \times 61\end{pmatrix} = \begin{pmatrix}210 - 122 \\ -84 + 183\end{pmatrix} = \begin{pmatrix}88 \\ 99\end{pmatrix}$$ 7. **Divide by determinant:** $$\begin{pmatrix}x \\ y\end{pmatrix} = \frac{1}{11} \begin{pmatrix}88 \\ 99\end{pmatrix} = \begin{pmatrix}\frac{88}{11} \\ \frac{99}{11}\end{pmatrix} = \begin{pmatrix}8 \\ 9\end{pmatrix}$$ 8. **Interpretation:** The price of one ream of A4 paper is $x = 8$ and the price of one ream of A3 paper is $y = 9$. **Note:** The values $x = -92$ and $y = 159$ given in the prompt are incorrect based on the system and calculations.