Solve for $x$ and $y$, $x \begin{bmatrix} 2 \\ 1 \end{bmatrix} + y \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatrix} = O$.
Answer & explanation
Correct answer: option 1
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
$2x + 3 \times 2 - 8 = 0$
$\Rightarrow 2x = 2 \Rightarrow x = 1$
$∴x = 1 \text{ and } y = 2$