Using the matrix method, the solution of the system of equations 2x -…

The correct answer is Option (3) → \(x = 3,\ y = 2\)

The system is \(AX = B\), where

    \( \displaystyle A = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix},\ X = \begin{bmatrix} x \\ y \end{bmatrix},\ B = \begin{bmatrix} 4 \\ 9 \end{bmatrix} \)

    \(|A| = 6 + 1 = 7 \ne 0\)

So the system has a unique solution \(X = A^{-1}B\).

    \( \displaystyle A^{-1} = \frac{1}{7}\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \)

    \( \displaystyle X = \frac{1}{7}\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\begin{bmatrix} 4 \\ 9 \end{bmatrix} = \frac{1}{7}\begin{bmatrix} 21 \\ 14 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \)

So x = 3 and y = 2. Check: \(2(3) - 2 = 4\) and \(3 + 3(2) = 9\).