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I do economics but am very poor at Math. I had a specific and perhaps silly question about the implicit function theorem, but will be grateful for an urgent response.

Suppose we have a function, U(x, y).

x and another variable z are linearly related so the function can also be specified as U(z,y) by substituting z for x.

It can be shown by using the implicit function theorem that y = f(x), and also separately that y = g(z) such that U(x,y)=0 and U(z,y)= 0 respectively.

Is it them possible to conclude that y = h(x,z) exists ?

Thank you in advance !