Suppose I have a complicated electric circuit which is composed exclusively of resistors and voltage and current sources, wired up together in a complicated way. The standard way to solve the circuit (by which I mean finding the voltage across, and current through, each circuit element) is to formulate Kirchhoff's laws for both current and voltage, and these will yield linear equations which enable one to solve for all the relevant quantities.
However, there are two problems with these laws:
There are too many of them. For example, in the simple circuit below, there are three different possible loops one can draw, but only two independent voltages. Similarly,
The equations are not all independent. In the circuit below, the current conservation equations for the two different nodes turn out to be exactly the same equation.
Fortunately, in real life, these problems happen to exactly cancel out, and one gets exactly the correct number of equations to solve the circuit. There are never too many contradicting constraints (the linear system is never overdetermined) and there are always enough equations to pin everything down (the linear system is never underdetermined).
Why is this? Is there a simple proof of this fact? What are the fundamental reasons for it?