Problem 1 - Algebra 1: Concepts and Skills

Question

Explain what it means for x and y to vary directly.

Step-by-Step Solution

Answer

Direct variation indicates a proportional relationship between two variables, x and y. If x increases or decreases, y will do the same correspondingly in a fixed ratio, denoted by the constant of proportionality 'k'. The relationship is represented as y = kx.

Step 1: Concept of Direct Variation

In mathematics, when two variables 'x' and 'y' vary directly, it signifies a specific kind of relationship between them. They are said to be directly proportional. In a direct variation, as one variable increases, the other variable also increases proportionally and vice versa.

Step 2: Proportionality Remark

If 'x' and 'y' are directly proportional, there exists a constant, 'k', such that y = kx. Here, 'k' is termed as the constant of variation or the constant of proportionality.

Step 3: Example of Direct Variation

For instance, if 'y' is directly proportional to 'x' with the constant 'k' as 3 then the equation is y = 3x. To illustrate, if you plug in 'x' as 2, then y will be 6, this demonstrates that as 'x' increases or decreases, 'y' does as well in the equal proportion.