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Here is a possible solution:

sage: var('beta o u y R')
(beta, o, u, y, R)
sage: S = (u+y)*(u+o)/(beta*o)
sage: S.subs(solve(R == beta*o/((u+y)*(u+o)), beta)).simplify_full()

Explanations: you have first to introduce R as a symbolic variable; this is what is done in the first line above. Then you express one of the other symbolic variables in terms of R; here we choose beta:

sage: solve(R == beta*o/((u+y)*(u+o)), beta)
[beta == (R*o*u + R*u^2 + (R*o + R*u)*y)/o]

This outcome is then used to perform the substitution in S via the function S.subs(...).