swing the balance idiom
使天平发生倾斜，起决定性作用
If something swings the balance, it is the thing that causes a particular situation to happen or a particular decision to be made instead of other situations or decisions that are possible.
This latest election promise might just swing the balance in the candidate's favour. 最近作出的这一选举承诺可能会对选民抉择起到决定性的作用。