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I have two systems of non-linear equations. The equations are continuously differentiable in the domain that I am interested in.

Let $x_1, y_1$ solve the following system of equations.

$f(x,y)=0\\ g(x,y)=0$

Let $x_2,y_2$ solve

$f(x,y)=0\\ g(x,y)=c$

Were $c>0$ is a constant. For simplicity and for starters we can assume that these solutions are unique. I have the functional forms of $f$ and $g$ and they are very long, painful and non-linear. I have some non-math intuition that the following are true and I would love some help and suggestions on the things I can try to do to prove these statements:

$x_2 > x_1 \\ y_2 > y_1 \\ x_2-x_1 > y_2-y_1 $

Thanks a lot,


Please feel free to make simplifying assumptions. For example, we can take $c>0$ to arbitrarily small for starters and see where that takes us.

I'm not sure if I can show this but suppose $f(x,y)$ and $g(x,y)$ satisfied the implicit function theorem then perhaps we could get some traction on this.

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@amWhy Thank you. I will drop the subscripts. – Amatya Nov 20 '12 at 22:56
Thanks, Amatya! I just wanted to be sure I understood the question. – amWhy Nov 20 '12 at 22:59
up vote 0 down vote accepted

The only change from the first system of equations to the second is a vertical displacement of the g function by -c units. The geometrical interpretation of the solutions of a system of equations is the points where the plotted graphs of all functions coincide. From the information you gave us, it is not possible to prove those identities because when displacing the g function it's cross points with the f function depend on the functional form.

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