What is the definition of a directly proportional relationship?

A directly proportional relationship between two variables means that as one variable increases or decreases, the other variable changes in a way that maintains a constant ratio. This can be expressed mathematically as ( y = kx ), where ( k ) is a constant. This indicates that for any two values of the variables, the ratio ( \frac{y}{x} ) remains equal to the constant ( k ).

For example, if you double the value of one variable, the other variable also doubles, maintaining that fixed ratio. This principle is fundamental in understanding linear relationships in mathematics and includes real-world applications in various fields, from physics to economics.

The other choices describe different types of relationships. For instance, stating that the product of the two variables is constant indicates a quadratic relationship rather than a direct proportionality. Similarly, mentioning that the sum of the variables remains unchanged refers to a linear relationship, while the assertion that the difference remains the same implies a fixed difference, indicating a constant relationship instead of a proportional one. Thus, the correct definition of direct proportionality is the one that accurately describes the ratio of the two variables as a constant value.