How can I solve $100+75x+225y=100+10\sqrt{5}x^{-1}+10\sqrt{5}y^{-1}$

Question

I find it hard to solve explicit for a variable when it looks something like this: $100+75x+225y=100+10\sqrt{5}x^{-1}+10\sqrt{5}y^{-1}$

Is there a trick or intuitive way of finding an expression of x using y?

Answer 1

HINT: you can multiply by$$xy\ne 0$$ then we get $$75x^2y+225xy^2=10\sqrt{5}y+10\sqrt{5}x$$ this can be solved for $x$ or $y$ We get $$y_1=\frac{-15 x^2-\sqrt{225 x^4+300 \sqrt{5} x^2+20}+2 \sqrt{5}}{90 x}$$ or $$y_2=\frac{-15 x^2+\sqrt{225 x^4+300 \sqrt{5} x^2+20}+2 \sqrt{5}}{90 x}$$

Answer 2

\begin{align} 100+75x+225y&=100+10\sqrt{5}x^{-1}+10\sqrt{5}y^{-1}\\ 75x + 225y&=10\sqrt5/x + 10\sqrt5/y\\ 15x + 45y&=2\sqrt5/x+2\sqrt5/y\\ 15x^2y+45y^2x&= 2\sqrt5y + 2\sqrt5x \end{align} Which can now be solved in terms of $x$ or $y$ using quadratic tricks

Answer 3

There is no trick involved. Since you assume that $x,y\neq 0$, multiplication by $xy$ gives an equation of the type $$ ax^2y+bxy^2+cx+dy=0. $$ This can be solved with "usual algebra", not tricky at all.