can I reuse an external hdd to make a new backup of another macbook?

Years ago I used an external hdd & TimeMachine to make a backup of my old macbook. That's history and I don't need the data anymore. Now I have another, newer Macbook (Air) that I should upgrade, and before that, backup. Can I just plug in this external hdd I've used before (with the backup of the old macbook still on it) and run TimeMachine to make a backup of my current Macbook?

Thanks for answering!