Assume that T(n) <= 10cn, where c is a constant.
Prove it by induction:
First, there is T(1) = O(1), satisfying that T(1) <= 10cn.
Assume that T(m)<=10cm for all m <= n-1
thus, for n, we have T(n) <= T(n/5) + T(7n/10) + cn
since n/5 <= n-1, 7n/10 <= n-1
For n, there is:
T(n) <= 10c * n/5 + 10c * 7n/10 + cn = 2cn + 7cn + cn = 10cn
Thus, T(n) <= 10cn for all n's. So that T(n) = O(n)