What happens to y if y varies directly as x, and x is halved?

Verified step by step guidance

1. Understand the concept of direct variation: If y varies directly as x, it means that y is equal to a constant k multiplied by x. This can be written as the equation \(y = k \times x\), where k is a constant.
2. Identify what happens when x is halved: If the original value of x is changed to \(\frac{x}{2}\), substitute this new value into the direct variation equation.
3. Write the new equation for y when x is halved: \(y_{new} = k \times \frac{x}{2}\).
4. Simplify the expression: \(y_{new} = \frac{kx}{2}\).
5. Compare the new y value to the original y value: Since the original y was \(y = kx\), the new y is exactly half of the original y, meaning y is halved when x is halved.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another, expressed as y = kx. If x changes, y changes proportionally, maintaining the constant ratio k.

Effect of Scaling the Independent Variable

When the independent variable x is scaled by a factor, the dependent variable y changes by the same factor in direct variation. For example, halving x results in y being halved as well.

Constant of Proportionality

The constant k in y = kx remains unchanged when x changes. It defines the fixed ratio between y and x, ensuring that y varies directly and proportionally with x.