Question

In a direct variation equation, how are the constant of variation and the slope related?

Short answer

In a direct variation equation, the constant of variation is directly equal to the slope. It represents the rate of change.

Step-by-step solution

Understanding direct variation

First, let's understand what direct variation means in mathematical terms. When two quantities x and y directly vary with each other, it means when one increases, so does the other, and vice versa. It is expressed in the form \(y = kx\), where 'k' is the constant of variation.

Understanding slope

In a linear equation, such as \(y = m x + c\), the slope, represented by 'm', measures the rate of change. It tells us how, for every unit change in x, the value of y changes

Comparing the two

From the two equations, it can be seen that in a direct variation equation, 'k' (the constant of variation), plays the same role as the slope does in a linear equation. There is no 'y-intercept' in a direct variation equation, which means the slope (or the rate of change) is the constant of variation. In mathematical terms, slope = k.