Implicit function theorem problem from Munkres Analysis on Manifolds text

Question

For the problem below, which is from Munkres' Analysis on Manifolds section on the implicit function theorem, I'm getting the following solution $\frac{\partial u}{\partial y} \neq \frac{18}{5}$, but I saw a solution giving the answer as $\frac{\partial u}{\partial y} \neq \frac{2}{5}$

My attempt:

I began by checking the conditions for both $G$, and $H$, where $det\big(\frac{\partial G(2,-1,1)}{\partial y} \big) \neq 0$ and the same for $H$

(1) $\det \big(\frac{\partial H(2,-1,1)}{\partial y} \big)=2\frac{\partial u}{\partial y}+9+3\frac{\partial u}{\partial y}=0$

(2) $\det \big(\frac{\partial G(2,-1,1)}{\partial y} \big)=f'(x,y)\frac{\partial u}{\partial y}+2u\frac{\partial u}{\partial y}=0$

and substituting (1) into (2) gives

$f'(x,y)\frac{\partial u}{\partial y}=\frac{18}{5}$

My thinking is that I was supposed to do something more with $f'(x,y)$ so that I could isolate $\frac{\partial u}{\partial y}$?

Thanks

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Answer 1

No, you have to look at the determinant of the $2\times 2$ matrix $\dfrac{\partial(G,H)}{\partial(x,u)}$ and see that it is nonzero.