To wear out, to vanquish.

1

a. 1848.  Either flax out your opponent, or give nature special fits in the undertaking.—Dow, Jun., ‘Patent Sermons,’ i. 54.

2

1866.  I think to flax for to beat [is American].—Lowell, Introd. to ‘Biglow Papers.’ (N.E.D.)

3

1903.  ‘Dialect Notes,’ ii. 351, quotes flaxed out as an Ohio term for worn out.

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