tip/turn the scales, to

tip (or turn) the scales (or balance)

(of a circumstance or event) be the deciding factor; make the critical difference.
See also: Scales, tip

tip the scales

1. To register weight (at a certain amount).
2. To offset the balance of a situation.
See also: Scales, tip

tip/turn the scales, to

To change the balance slightly so as to favor one side. The idea was expressed as long ago as Aristotle’s Politics (ca. 340 b.c.), in which he wrote “The addition of the middle class turns the scale and prevents either of the extremes from being dominant” (Benjamin Jowett translation, ca. 1875). Jowett’s wording may have been influenced by William Wordsworth’s “Ode” (1815): “Thy presence turns the scale of doubtful fight, / Tremendous God of battles, Lord of Hosts!”
See also: tip, turn