tip the balance
Also, tip the scales; turn the scale. Offset the balance and thereby favor one side or precipitate an action. For example, He felt that affirmative action had tipped the balance slightly in favor of minority groups, or New high-tech weapons definitely tipped the scales in the Gulf War, or Just one more mistake will turn the scale against them. Shakespeare used turn the scale literally in Measure for Measure (4:2): "You weigh equally; a feather will turn the scale." The idioms with tip are much younger, dating from the first half of the 1900s.