From Lenz's Law to Faradays Law Relation Is there a way to mathematically describe how one gets to the minus in Faraday's Law from Lenz's Law?
 A: Faraday's Laws gives the relation that the electomotive firce or emf is directly proportional to the rate of change of magnetic flux, i.e. $$E \propto \dfrac {d\phi}{dt} $$ It does not give an equation.
Lenz's law puts into effect the law of conservation of energy to say that the EMF is equal to negative of rate of change of flux, i.e. $$ E = - \dfrac {d\phi}{dt} $$
So, both are related. Faraday's law gives the relation, while Lenz's Law puts it into an equation.
Description to get to Lenz's Law is the law of conservation of energy. It is difficult to write that out. Think of a magnetic and a solenoid. If the magnetic is moved towards the solenoid, then a pole, same as that of the one of the magnet facing the solenoid develops on the solenoid and the current flows accordingly. This is done to create a repulsion and prevent the magnetic from moving,  thereby conserving momentum and energy.
Opposite happens when the magnetic is moved away. An opposite Pole is developed which tries to attract and prevent the motion of the magnet. 
The diagram may partly help you, though it is drawn badly:

A: The flux through the coil is in one direction or the other, whereas the emf doesn't have a direction in space, but a sense around the coil. It might therefore seem hard to make sense of either a + or a - sign in the equation relating $\varepsilon$ and $\frac{d \Phi}{dt}.$ The key to understanding what it means is a sign convention: let the z direction be one of the two possible directions (it doesn't matter which) along the coil axis, then a positive emf is one whose rotation sense would advance an ordinary right-handed screw in the +z direction. Thus the minus sign in $\varepsilon = -\frac{d \Phi}{dt}$ is telling us that the rate of change of flux producing the emf is in the opposite direction from that given by the right hand screw rule (with the screw being turned in the sense of the emf).
This answer is not attempting to explain how Lenz's law arises, but how the minus sign in $\varepsilon = -\frac{d \Phi}{dt}$ represents Lenz's law.
