Calculating inverse function with 2 variables

Question

$f: R^{2}\mapsto R^{2}$

$(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$

I should calculate inverse function of $f$ in point $(3,1)$.

I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x and y from it, because everything is mixed?

$u=x^{2}-4y^{2}+x$

$v=-xy+3y$

Answer 1

Hint: You need to solve for x and y simultaneously the two equations $x^2-4y^2+x=3,\ -xy+3y=1.$ One start would be to solve the second one for $y$ [easy since second is linear in $y$] and then plug that into the first one to get a single equation in just $x.$ This resulting $x$ equation may or may not be easily solved, but that seems the way to approach finding the desired inverse.

Note that this is not the same as plugging $x=3,y=1$ into the two formulas and getting $(8,0).$ Doing that amounts to finding the forward image, or simply the image, of the point $(3,1),$ rather than the inverse image of that point.