Question

$left { {{sqrt{x/y} - sqrt{y/x} = 3/2} atop {x+y+xy=9}} right$

Answer 1

$frac{x}{y}=a\\ sqrt{a} - sqrt{frac{1}{a}}=frac{3}{2}\ sqrt{a}-frac{1}{sqrt{a}}=frac{3}{2}\ frac{a-1}{sqrt{a}}=frac{3}{2}\ 2a-2=3sqrt{a}\ 4a^2-8a+4=9a\ 4a^2-17a+4=0\ D=15^2\ a=frac{17+15}{8}=4\ a=frac{17-15}{8}=frac{1}{4}\ x=4y\ y=4x\\ 5y+4y^2=9\ 5x+4x^2=9\\ 4y^2+5y-9=0\ D=25+4*4*9=13^2\ y=frac{-5+13}{8}=1\ y=frac{-5-13}{8}=-frac{9}{4}\ x=4\ x=-9$