Question: The equation $$7xyz=2x^2+y^2+3z^2+7$$ implicitly defines z as a function of x and y in the neighborhood of the point where $x=2, y=1$ and $z=2$ . Find ∂z/∂x and ∂z/∂y at this point.
Attempt at solution:
So I rearranged the equation to give me $$0=2x^2+y^2+3z^2+7-7xyz$$, then took the partial derivatives of ∂F/∂x/∂F/∂z, which got me $$\frac{4x-7yz}{6z-7xy}$$ and to get ∂z/∂y, I took ∂F/∂y / ∂F/∂z which got me $$\frac{2y-7xz}{6z-7xy}$$ Plugging in x, y, and z got me ∂z/∂x = 4, and ∂z/∂y = 13, but it's wrong. Where did I go wrong?