# Signing partial derivatives without using Cramer's rule in a system

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.