Question

Find the values of $xandy$ if

$\left|\begin{array}{cc}x& y\\ 3& 2\end{array}\right|=6and\left|\begin{array}{cc}5& x\\ 4& y\end{array}\right|=-8$

Answer 1

$\left|\begin{array}{cc}x& y\\ 3& 2\end{array}\right|=6$

$\Rightarrow 2x-3y=6$

$\Rightarrow 2x=6+3y$

$\Rightarrow x=\frac{6+3y}{2}=3+\frac{3y}{2}$

$\left|\begin{array}{cc}5& x\\ 4& y\end{array}\right|=-8$

$\Rightarrow 5y-4x=-8$ or $4x-5y=8$

Substituting $x=3+\frac{3y}{2}$ in $4x-5y=8$

$4\times (3+\frac{3y}{2})-5y=8$

$\Rightarrow 2\times (6+3y)-5y=8$

$\Rightarrow 12+6y-5y=8$

$\Rightarrow y=8-12=-4$

$\therefore y=-4$

Substituting $y=-4$ in $x=3+\frac{3y}{2}$ we get

$x=3+\frac{3\times -4}{2}=3-6=-3$

$\therefore x=-3,y=-4$