Solve for $x$ and $y$, $x \begin{bmatrix} 2 \\ 1 \end{bmatrix} + y \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatrix} = O$.

$x=1,y=2$

The correct answer is Option (1) → $x=1,y=2$ ##

We have,

$x \begin{bmatrix} 2 \\ 1 \end{bmatrix} + y \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatrix} = O$

$\Rightarrow \begin{bmatrix} 2x \\ x \end{bmatrix} + \begin{bmatrix} 3y \\ 5y \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatrix} = O$

$\Rightarrow \begin{bmatrix} 2x + 3y - 8 \\ x + 5y - 11 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

On comparing the corresponding elements of matrices, we get

$2x + 3y - 8 = 0$ ---(i)

On multiplying Eq. (i) by 2, we get

$4x + 6y = 16$ ---(ii)

and $x + 5y - 11 = 0$ ---(iii)

On multiplying Eq. (iii) by 4, we get

$4x + 20y = 44$ ---(iv)

On subtracting Eq. (ii) from Eq. (iv), we get

$14y = 28 \Rightarrow y = 2$

On putting the value of $y = 2$ in Eq. (i), we get

$2x + 3 \times 2 - 8 = 0$

$\Rightarrow 2x = 2 \Rightarrow x = 1$

$∴x = 1 \text{ and } y = 2$