Simpler equation for perfectly elastic collisions.

Perfectly elastic collisions problems usually involve calculating the final velocities of two masses from their initial momenta. Trying to derive such formula I got a different result, a shorter formula to solve the same problem:
Take two masses a and b with their respective initial volocities;
First I assumed the velocity of the center of mass to be constant;
vc=const.
Then I moved my referential to the mass a. In this referential I assumed that the absolute value of the relative velocites between the mass a and the center of mass to be also constant.
What I imagined what more or less like this:
Before the collision I would see the center of mass move towards my referential with a velocity "via-vc". After the collision I would see the center of mass move in the opposite direction with the same speed;
Based on this what I got was:
|via-vc|=|vfa-vc|=const.
via-vc=vc-vfa
via+vfa=2*vc
That's it. The oddity is that it uses a rather faster thought, works perfectly and I've never seen before.
Now you can solve collisions problems with a quicker equation :)

$$v_1 - v_2 = v_2' - v_1'$$​