I’m trying to solve a problem involving the collision of 2 particles. One particle has a mass of 8 and is moving in the +x direction at (15/17)c, and the other has a mass of 12 and is moving in the –x direction at (5/13)c. I wind up with 2 equations in 2 unknowns but I can’t figure out how to do the algebra.

It’s an elastic collision and I’m looking for the final velocities of each particle. I’m using the conservation of energy (30) of the two particles and the conservation of momentum of the two particles to come up with the 2 equations in 2 unknowns. The equation I posted is the one for the conservation of energy.

I can consider one of the particles to be at rest and use the relativistic addition formula to figure out velocity of the other particle but I don’t know if the non relativistic equations still hold

Yeah, you could consider one at rest and do everything in that frame, but you should still wind up with a similar formula. You could also put everything in the zero-momentum frame, which would make the final momentum equation simpler.

What you’re trying to do in the OP, get v1 in terms of v2, and then substituting it into the other equation, is what I like to call “thinking inside the box”. Just because it’s the most straightforward way to do it doesn’t mean that’s the way to go. I would suggest looking for another way to eliminate v1 or v2 from the equations.

(Now that I’ve given my kingly advice, watch this be one time when thinking inside the box is the way to go.)

The center-of-momentum frame is definitely the way to go, for a simple solution. For a two-body elastic collision in the CM frame, the particles must leave with the same speeds they came in with; the only remaining parameter is the direction they leave in. (Proof is an exercise for the student.:)) It looks like this is a one-dimensional problem, so you know that too. Then all you have to do is the transformations to and from this frame, which require relativistic velocity addition.