I have a system of two equations and two variables (x,y) and a parameter (a) :
$f(x,y;a) = 0$
$g(x,y;a) = 0$

I know that using Cramer's rule, I can find the expressions of $x_a=\frac{\partial x}{\partial a}$ and $y_a = \frac{\partial y}{\partial a}$. I am assuming here that the Jacobian of this system is invertible.

After taking the partial derivative with respect to $a$ on both equations and setting the left hand sides equal to each other, I know I can (assuming all inverted terms are invertible) write,

$ x_a= \frac{(g_a-f_a)}{(f_x-g_x)} + \frac{(g_y-f_y)}{(f_x-g_x)}y_a$

My question(s):

Given that I know the sign of the first term on the RHS and the sign of $(f_x-g_x)$,
(i) can I make statements like: "if $(g_y-f_y)$ is of a particular sign then $sign(x_a) = sign (y_a)$" ?
(ii) will such statements be necessarily true?
(iii) or will they only be sufficient?

I really do not want to actually compute the partials or use Cramer's rule.

For example:

Suppose $g_a-f_a = 0$, and $f_x-g_x > 0$. $^*$
(i) Is the statement $sign(x_a) = sign(y_a)$ if $(g_y-f_y)>0$ correct?
(ii) Is it necessarily true?
(iii) Is it only sufficient?

Note: In the actual problem I am dealing with, the partials of $f$ and $g$ with respect to the variables will sometimes contain these variables. So I understand that signing groups of partials (as I have done above in $(*)$) will involve imposing restrictions on values these variables (or other parameters in the system) can take.


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