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]
(..) Oh, there's more to being king than-- getting your way all the time. There's more? Simba. Everything you see exists together, in a delicate balance. As king, you need to understand that balance, and respect all the creatures from the crawling ant to the leaping antelope. (..)1994 The Lion King