Four equations, five unknowns, all linear. This isn't particularly difficult, just tedious.

Because there are more unknowns than equations, your solution should itself be an equation expressing one of the unknowns as a linear function of the other four -- omega = a * beta + b * gamma + c * phi + d * mu, where a, b, c, and d are found by chugging away using your favorite linear solution technique (Gaussian elimination, perhaps.)

I leave the actual solution as an exercise for the reader ;)