Once again with proper formatting and some tiny comments:
eq1 = k*X''[Y] + Bi*(Z[Y] - X[Y]) == Subscript[U, p]/Subscript[U, m];
eq2 = Z''[Y] - Bi*(Z[Y] - X[Y]) == 0;
Solve eq2 first:
sol2 = Solve[eq2, Z[Y]]
Insert solution of eq2 into eq1:
eq3 = Simplify[eq1 /. Flatten[sol2]]
Flatten is used to remove the extra {}. Final results is:
(* k (X^\[Prime]\[Prime])[Y] + (Z^\[Prime]\[Prime])[
Y] == Subscript[U, p]/Subscript[U, m] *)