The phrase "hinge primarily on" is correct and commonly used in written English.
It means that something is largely dependent or based on something else. Example: The success of the project will hinge primarily on the support and cooperation of the team members.
Mr. Gore's hopes of overcoming Mr. Bush's slim margin in Florida now hinge primarily on the Florida Supreme Court's overturning this week's ruling by Judge N. Sanders Sauls and ordering a manual count of some 9,000 punch-card ballots in Miami-Dade County that did not register a presidential choice when counted by machine.
Outcomes of surgical aortic valvuloplasty hinge primarily on optimal leaflet design.
This coming together of past and future at the moment of disclosure of a diagnosis, combined with an expressed notion that the consequentiality of the diagnosis trumps the urgency of diagnosis, seemed to hinge primarily on the unpredictability associated with dementia.
Drug treatment of malaria currently hinges primarily on one single type of drug, the artemisinin derivates.
Presidential popularity hinges primarily on several factors.
Officials at Harvard and Princeton said Dr. West's decision had not hinged primarily on money.
Mr. Graham's conviction hinged primarily on the testimony of a single witness, who testified she saw him from 30 to 40 feet away through her car windshield.
Ms. Caldwell, the Edmunds.com analyst, said G.M.'s ability to restore its image hinges primarily on building better vehicles, which it has been doing.
The debate hinges primarily on whether the flaws in the studies supporting mammography are so serious that they invalidate the results or whether they are of lesser magnitude and leave the bottom line intact.
Resistance to mechanical disruption hinges primarily on homeostasis of the dermal collagen fibril architecture, which is regulated, at least in part, by members of the small leucine-rich proteoglycan (SLRP) family.
He said he had told EU leaders the Brexit referendum had hinged primarily on the immigration question because of the EU's insistence on the freedom of movement for its citizens.
