Master Theorem Usage I know how to use the master Theorem for a general formula e.g. $T(n) = a \cdot T(\frac{n}{b}) + f(n)$.

I saw some books suggest that we can use the mastere theorem for formulas like:
$T(n) = T(\frac{n}{5}) + T(\frac{7n}{10}) + O(n)$

How is that possible?

My guess: We use it separately. Once for $T(n) = T(\frac{n}{5}) + O(n)$ and once for $T(n) = T(\frac{7n}{10}) + O(n)$ and join between.

Thanks!
You can use the idea of the Master Theorem and apply it to these occurrences. That is what I have done here, I would recommend looking at that answer because it will answer your question.

Basically we can look at the fractional terms inside the recurrence calls. If the sum to less than $n$, then the "conquer" part of "divide-and-conquer" will dominate the "divide" part. 

For your example we have the fractions: $\frac{n}{5} + \frac{7n}{10} = \frac{9n}{10} &lt; n$. Therefore the $O(n)$ term will dominate and the overall time complexity will be $\Theta(n)$. This is an idea of the Master Theorem, figuring out which dominates: "divide" or "conquer". Again, I highly suggest looking at this answer for a more in-depth proof.