Definition
When a responsibility or decision lies with someone, they bear the responsibility for it.
由（某人）负责
The final decision was authorized to the manager, so all responsibility rests with him.
最终的决定权在经理，因此所有的责任都由他承担。
The success or failure of this project ultimately rests with you.
这个项目的成功或失败最终由你负责。
I can't decide the itinerary for the trip; it has to rest with you.
我无法决定旅行的行程，这必须由你来决定。