Check $x_0$ $y_0$ in $U'$ and $U''$ that solves the equation

Question

here is something i dont understand

Check for the equation in $ x \in \mathbb{R}^{k} $ , $ y \in \mathbb{R}^{n} $ if there exists an envinronment $U', U''$ of $x_{0}=0, y_{0}=0$ where exists a $y$ to every $x$, that solves the equation:

$\cos(x_{4})*\sin(y)^{4}+(x_{2}-7)y^{2}=0$ for $k=4$ and $n=1$

As far as i heard i have to do the jacobian matrix and check if the matrix can be inverted and if the function is injective. But i dont know if this is right and how to do it. The equation has many variables. Is this a $R^{n+k} \rightarrow R^1$? I am asking because the jacobian matrix would be a $R^{n+k} \times R^1$ then. But if this is so, i cannot check if the matrix can be inverted because it would not be quadratic. Am i right? Can you help me?

Thank you in advance for any help!

Answer 1

You need to use the implicit function theorem. Take a look at this webpage Implicit Function Theorem. You cannot invert the Jacobian.