<s>[INST] A fraction problem $$a = x + \frac1x \\b =y + \frac1y \\ c = xy +\frac1{xy} $$ 
Express $c$ in terms of $a$ and $b$
 [/INST] Multiply all things by the LCD to remove fractions.
$$ax=x^2+1$$
$$by=y^2+1$$
$$cxy=x^2y^2+1$$
Multiply the first two equations to get
$$abxy=x^2y^2+1+x^2+y^2=c+x^2+y^2$$
Adding the first two equations, we get
$$ax+by=x^2+y^2+2$$
Combining these two lines,
$$abxy=c+ax+by-2$$
At this point, I'd just go back and solve for $x,y$ using the quadratic formula,
$$c+a\left(\frac{a\pm\sqrt{a^2-4}}2\right)+b\left(\frac{b\pm\sqrt{b^2-4}}2\right)-2=ab\left(\frac{a\pm\sqrt{a^2-4}}2\right)\left(\frac{b\pm\sqrt{b^2-4}}2\right)$$
Not the most beautiful thing in the world, but at least you don't have to square the thing or anything messy like that.
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