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She went to the beauty salon to have her hair done.
Literal
She [topic-は] hair [object-を] tied [have-someone-do-てもらう] [in order to-ために] beauty-salon [to-に] went.
~ために expresses purpose with a verb in dictionary form — 'in order to.' Note that ~ために on a noun would use の (e.g., 健康のために 'for the sake of health'); after a verb, it attaches directly. 結う ('to do up, tie [hair]') is a slightly old-fashioned, specifically-hair verb; modern speech often prefers セットする.