How do I derive the linear momentum formula for a elastic collision

So I have M(1)V(1)i + M(2)V(2)i = M(1)V(1)f + M(2)V(2)f
How do I make the formula to get these:

V(1)f = [(M(1) - M(2)) / (M(1) + M(2))]V(1)i + [(2M(2)) / (M(1) + M(2))]V(2)i
and
V(2)f = [(2M(1)) / (M(1) + M(2))]V(1)i + [(M(2) - M(1)) / (M(1) + M(2))]V(2)i

You have to have a second equation which shows kinetic energy is unchanged (an elastic collision). Using your symbols:
½M(1)(v1)² + ½M(2)(v2)² = ½M(1)(v1f)² + ½M(2)(v2f)² (the½s will cancel).

You solve the 2 simultaneous equation (momentum and kinetic energy) to find the 2 unknowns V(1)f and V(2)f.

The algebra is very long and messy and there is a better way. If you are in the centre-of-mass frame of reference the velocities are reversed. This simplifies the algebra. Take a look at the link.