Solve the system of equations: $2x + 5y = 1$, $3x + 2y = 7$.

$x = 3, y = -1$

The correct answer is Option (1) → $x = 3, y = -1$ ##

The system of equations can be written in the form $AX = B$, where

$A = \begin{bmatrix} 2 & 5 \\ 3 & 2 \end{bmatrix}, X = \begin{bmatrix} x \\ y \end{bmatrix} \text{ and } B = \begin{bmatrix} 1 \\ 7 \end{bmatrix} \text{}$

Now, $|A| = -11 \neq 0$. Hence, $A$ is a nonsingular matrix and so has a unique solution.

Note that $A^{-1} = -\frac{1}{11} \begin{bmatrix} 2 & -5 \\ -3 & 2 \end{bmatrix} \text{}$

Therefore $X = A^{-1}B = -\frac{1}{11} \begin{bmatrix} 2 & -5 \\ -3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 7 \end{bmatrix} \text{}$

i.e., $\begin{bmatrix} x \\ y \end{bmatrix} = -\frac{1}{11} \begin{bmatrix} -33 \\ 11 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \text{}$

Hence $x = 3, y = -1$.