How to express a variable from the system not manually?

Question

Mathematics

There is a system of five equations with five unknowns.

M*omega^2*R = T*(sin(alpha)+sin(beta)), 
M*g = T*(cos(alpha)-cos(beta)),  
m*g = T*cos(beta), 
m*omega^2*R = T*sin(beta), 
L = R*(1/sin(alpha) + 1/sin(beta))+r/sin(beta)

It is necessary to express, for example, alpha through M, m, omega, g, L. Manually, it's easy to see, it's too hard to do. How to do it programmatically or somehow?
Wolfram alpha tried to use, did not find the right one. I don't know anything else.

Answer 1

First, you didn't specify what is unknown.
Secondly, this system is solved elementarily.
You asked to express alpha. To do this, it is enough to find cos(beta) from Equation 3:
cos(beta) = mg/T
and substitute it into Equation 2, from where to express alpha:
cos(alpha) = (M+m)g/T
Well, that's it. The rest is also elementary, at the level of the 9th grade of the school.