Question

Solve for $x$ and $y$, using cross multiplication method
$\frac{b}{a}x+\frac{a}{b}\gamma =a^2+b^2,$x+y=2ab

Answer 1

$\frac{b}{a}x+\frac{a}{b}y=a^2+b^2⟶\left(I\right)$

$x+y=2ab⟶\left(II\right)$

Using Cross multiplication

If, $a_{1}x+b_{1}xy+c_1=0$

$a_{2}x+b_{2}y+c_2=0$

$x=\frac{b_1{c}_2-b_2{c}_1}{a_1{b}_2-a_2{b}_1}$ & $y=\frac{c_1{a}_2-c_2{a}_1}{a_1{b}_2-a_2{b}_1}$

$x=\frac{-2ab\times \frac{a}{b}-(-(a^2+b^2))}{\frac{b}{a}-\frac{a}{b}}$

$=\frac{-2a^2+a^2+b^2}{\frac{b^2-a^2}{ab}}$

$=ab\frac{(b^2-a^2)}{(b^2-a^2)}=ab$

So, $y=ab$ from $\left(I\right)$

$x=ab$ & $y=ab$