Two People Pushing Off of Eachother, Newton's Third Law, and Unbalanced Force Different versions of this question have come up all over the internet. Usually it deals with tension in a rope or two people pushing on each other with the same force. I am trying to understand 2 people pushing each other with different forces.  
Say two people of equal mass are standing on a frictionless surface, touching palm to palm, and they push off of each other at the same time with different amounts of force. Say the person on the left pushes with 100 N and the person on the right pushes with 70N. What happens to this system? It seems like there should be a net force of 30 N to the right, but at the same time, each person pushing should feel the same force pushing back as per Newton's 3rd Law. 
If I push on you with 100 N then I also feel the 100 N from the force pair and we slide apart each having been accelerated by 100 N. If you push on me with 70 N then I feel the 70 N force and I exert 70 N back on you, we slide apart each having felt 70 N this time. But what happens when we both push with a 100 N and 70 N respectively?  
Also, I have seen elsewhere on this site that if 2 people were to push on each other with the same force under conditions like those in the scenario above, they each feel the same force but fly apart at double the final velocity because the same force has been applied for twice the distance. Is this actually the case?
 A: The human body is terrible laboratory for Gedanken experiments, and this problem proves it. What exactly does pushing with 70N mean? Is it active? Is it passive? Nothing feels correct.
This is why Newton's Laws are quite often imagined with masses and springs, which is the only way to save this problem from comments explaining the answer is insufficient or feels wrong.
So: Imagine each person pushes not with a hand, but with a giant spring--a giant calibrated spring that measures force. If person A pushes with 100 N and person B does nothing, both their springs compress to read 100 N. (This is passive pushing for B). Now B fights back and actively adds 70 N to his spring: both springs compress to read 170 N.
Equal and opposite. Always.
A: 
Say the person on the left pushes with 100 N and the person on the right pushes with 70N. What happens to this system?

This is not possible. They can only push on each other with the same force. 
Suppose the person on the left is a weightlifter and capable of pushing very forcefully, and suppose that the person on the right is unconscious and not capable of exerting any force voluntarily. Even in such an asymmetrical circumstance the force will be the same. If the person on the right is rigid then the person on the left can generate their full force and the natural rigidity of the person on the right will return the equal and opposite force without effort. If the person on the right is limp then the person on the left cannot generate much force at all since the limp person will deform under minimal force. Either way the force is automatically guaranteed to be equal and opposite. 
A: The problem here stems from understanding closed systems. 
The center of mass of a closed system (no external forces) does not move. This comes from the fact that Newton's third law states that any internal force applied will be canceled by another opposite and equal internal force. 
The forces you and your friend are applying on each other are internal forces. The argument with 30 N that you made only applies if some external third person pushed your friend 100 N and then you 70 N. Those are forces outside the system where the equal and opposite force was applied to an object outside your system (the third person) causing a net change to your system. Because the forces in your description comes from within what you define as your system, the center of mass doesn't change i.e. no net force. Hope that helps.
