Final answer:
The Urubamba River flows through the Sacred Valley of the Inca and joins the Ucayali River in Peru, playing an important role in Inca history and agriculture.
Step-by-step explanation:
The river that flows through the Sacred Valley of the Inca and empties into the Ucayali River in Peru is the Urubamba River. This river runs through the heart of the once expansive Inca Empire, carving its path through the Andean landscape and acting as a vital water source for the ancient culture.
Along its course, the Urubamba River passes by the iconic Inca ruins of Machu Picchu, facilitating agriculture in the region through irrigation, and serving as a spiritual element in the Andean cosmology.