He's trying to keep his work life and his family life 'in balance'. = He's trying to keep his work life 'in balance with' his family life. [=he's trying to give a proper of amount of time and attention to both his work life and his family life]
The discount 'balances out' the shipping charge. = The discount and the shipping charge 'balance' each other 'out'. [=the shipping charge and the discount together equal zero]
He needs to achieve/create/strike a better 'balance' between his work life and his family life. [=he needs to spend less time at work and more time with his family]
(..) The murderer... You think the murderer has the phone? Maybe she...left it when she left her case. Maybe he took it from her for some reason. Either way, the balance of probability is the murderer has her phone. Sorry...what are we doing? Did I just text a murderer? What good will that do? (..)2010 TvShow: Sherlock Title: A Study in Pink Season: 1 Episode 1