I think you are worrying about some magic algorithm, when simple linear algebra will suffice. Yes. LINEAR algebra.
Each equation is of the same form
Ri = X_i*X_j*H_k
TAKE THE LOG. Then you have
log(R_i) - log(H_k) = log(X_i) + log(X_j)
So your equations are LINEAR in the log of the unknowns. Solve the linear system for the unknowns. Again, this is linear algebra.
You will find the linear system is singular, at least as you wrote the equations, so no solution will exist unless something rare happens with the constant terms. And if that does happen, then infinitely many solutions will eist.