I will start with an example to make things clear and avoid confusion :
Take all $x>0$ and $$\exp(x) = \sum_{-1<n} a_n x^n$$
Now finding $a_n$ can be described as an infinite linear system of equations :
$$a_0 + a_1 1 + a_2 1 + ... = e$$ $$a_0 + a_1 2 + a_2 4 + ... = e^2$$ $$...$$
and for every $x>0$
$$a_0 + a_1 x + a_2 x^2 + ... = e^x$$
So we have a countable number of variables and an uncountable number of equations.
Notice there is only one unique solution set $a_n$ here.
This leads to :
Conjecture :
An infinite linear system of equations with an uncountable number $A$ of equations, but a countable number $B$ of variables has at most countable ordinal $C$ * Linear independant * solutions, where $C < B$ and Linear independant means not a COUNTABLE LINEAR combination of others.
Notice that (in contrast) a countable number of (linear) equations with a countable number of variables CAN have an uncountable number of solutions, who are linear independant.
For instance the (formal) taylor series solution to
$f(x) = f(\exp(x))$
has an uncountable number of solutions and it is equivalent to a system of linear equations with a countable number of variables (the taylor coefficients).
Explicitly: $$t_i = \sum_n \frac{t_n n^i}{n!}$$
And examples without calculus, dynamics or functional equations exist as well.