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  4. If [-2&5 3&-1][x y] =[1&2 3&4][3 -1] then (x,y) is

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Q. If $\begin{bmatrix}-2&5\\ 3&-1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} =\begin{bmatrix}1&2\\ 3&4\end{bmatrix}\begin{bmatrix}3\\ -1\end{bmatrix}$ then $(x,y)$ is

COMEDKCOMEDK 2014Matrices

Solution:

$\begin{vmatrix}-2&5\\ 3&-1\end{vmatrix}\begin{vmatrix}x\\ y\end{vmatrix} =\begin{vmatrix}1&2\\ 3&4\end{vmatrix}\begin{vmatrix}3\\ -1\end{vmatrix} $
$\Rightarrow \begin{vmatrix}-2x+5y\\ 3x-y\end{vmatrix}=\begin{vmatrix}1\\ 5\end{vmatrix}$
$\Rightarrow \ - 2x + 5y = 1, 3x - y = 5$
or $y = 3x - 5 $
$\Rightarrow \ - 2x + 5 (3x - 5) = 1$
$\Rightarrow -2x + 15x - 25 = 1$
$\Rightarrow \ 13x = 26 \Rightarrow x = 2$
Substituting $x = 2$ in (i), we get
$y= 6 - 5 = 1$
Hence, $(x, y) = (2, 1)$