What describes the relationship between two variables in which their product is constant?

The relationship where the product of two variables remains constant is known as inverse variation. In this scenario, as one variable increases, the other must decrease in such a way that their product remains the same. This can be expressed mathematically as ( x \cdot y = k ), where ( k ) is a constant. For example, if one variable is doubled, the other variable must be halved to keep the product constant.

This concept contrasts with direct variation, where as one variable increases, the other also increases proportionally; in proportional relationships, the ratio between the two variables remains constant rather than their product. A linear relationship, on the other hand, typically represents a constant rate of change, which is not applicable in this case where the product, not the ratio or sum, is the focus of the relationship. Thus, the correct identification of the relationship involving a constant product is indeed inverse variation.