Why do the Tupi-Guaranis use the slash and burn method?
The land was not naturally fertile, so they had to use the slash and burn method of farming. The Tupi people were indigenous peoples in Brazil.
They didn't slash and burn their crops. They cut down forests(=slash) and set fire to the rest(=burn). This freed up surface area for farming, and the ashes made the ground fertile. Trouble was, they didn't use fertilizers or crop rotation. When they didn't get good yields from their lands they'd simply slash and burn some more. Old farm land was left almost empty of nutrients, with high risk of erosion and poor chances for reforestation.
What best explains why subsistence farmers continue to use slash-and-burn techniques in the Amazon River Basin?
The Soil itself was volcanic. The land was cleared using a technique somewhat similar to slash and burn except it was not exactly that. Instead it left charcoal on the surface. Unlike slash and burn, leaving charcoal on the surface left the soil fertile for a number of years. The volcanic soil was alkaline. The charcoal was acidic. That way the land was close to the correct pH for crops.