What did going to the center of mass velocity frame do for us? It told us the two balls reverse their velocties in that particular frame. But we want to know what happens in our original frame. Can we express what we just learned in a way that'll help us solve our original problem?
If we ask what happens to the difference in velocities before and after the collision, that'll tell us something independent of reference frame. The difference in the velocity of the two balls is always the same irrespective of frame. So what happens to the difference in velocities?
In the center of mass frame we see from eqn. 1.57
that 
.
But as was just stated, this is true in our original frame
That is, the difference in velocities just reverse sign after the collision. You could also say that for any one dimensional elastic collision relative velocity of approach equals the relative velocity of recession. This is true in any reference frame. This equation is nice because it's linear. Its key to figuring out problems with elastic collisions.
So we now have two linear equations that we have to solve
Multiply the second equation by 
, and add it to the first equation
to eliminate 
Solving for 
We can get most easily by interchanging the indices 1 and 2
Let's do a couple of examples now.
