Consider the system of non linear equations $$\begin{cases}x^2y^3+x^3y^2+x^5y+1=a \\ xy^2-2x^2y^4+3x^3y=b\end{cases}$$
How can I prove that for a $(a,b)$ close to $(4,2)$ there is a unique solution $x=f(a,b)$, $y=g(a,b)$ close to $(1,1)$?
Note that when $a=4,b=2$ the system has the solution $x=1,y=1$