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]
(..) He made a different call. And what will you do if I vow to spare him? Not let you out. No, but I like this. Your world in the balance, and you bargain for one man. Regimes fall every day. I tend not to weep over that. I'm Russian. Or I was. (..)2012 The Avengers