I am given a system of equations:
$F_1(t,x,y)=3x^3+2y^5-5t^7$
$F_2(t,x,y)=2x^3-y^5-t^7$
The task is to show that the system is locally uniquely resolvable in the point $(t,x,y)=(0,0,0)$ in respect to the variables $x,y$.
So obviously the implicit function theorem cannot be used (since the matrix $D_2$ isn't invertible at the given point). But if I substitute $a=x^3,b=y^5$ the matrix $D_2$ becomes:
\begin{pmatrix} 3 & 2 \\ 2 & -1 \end{pmatrix}
and is invertible as such so now I can apply the implicit function theorem. Since $x^3,y^5$ are injective and continuous functions the system is locally uniquely resolvable in $x$ and $y$, because it is resolvable in $a,b$, which follows from the theorem.
Is this a correct way to solve this and if not where would this fail?