Variables $x$ and $y$ satisfy the following system of equations
$f_1 (x,y,a) = 0$
$f_2 (x,y,a) = 0$
where $a$ is just a fixed parameter term. I would like to know how $x$ and $y$ changes with $a$ while satisfying the system. I do not know the closed form expressions for $x$ and $y$ , so I cannot simply find them and take partials wrt $a$. I know one can do that if there is just one equation by taking a total differential, but I am stuck when it is a system of equations. Any general guidance toward any existing method that tackles this problem is greatly appreciated.
Total differentials on each equation individually work fine to give you a pair of coupled DEs: \begin{eqnarray} \frac{\partial f_1}{\partial x}x' + \frac{\partial f_1}{\partial y}y' + \frac{\partial f_1}{\partial a} = 0\\ \frac{\partial f_2}{\partial x}x' + \frac{\partial f_2}{\partial y}y' + \frac{\partial f_2}{\partial a} = 0 \end{eqnarray} If you feel like you want the $x'$ and $y'$ isolated, some linear algebra gives $$ \begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}\partial_x f_1 & \partial_y f_1\\\partial_x f_2 & \partial_y f_2\end{bmatrix}^{-1}\begin{bmatrix}\partial_a f_1\\\partial_a f_2\end{bmatrix}. $$ The partial derivatives of $f_1$ and $f_2$ as well as the matrix inverse should be easy enough to compute, and hopefully the resulting pair of equations isn't some godawful mess.