Answer each question. 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 the constant of proportionality.
2. Express the original relationship: Initially, we have \(y = k \times x\). This means y depends on x multiplied by some constant k.
3. Consider the new value of x when it is halved: If x is halved, the new value of x becomes \(\frac{x}{2}\).
4. Substitute the halved x into the direct variation equation: Replace x with \(\frac{x}{2}\) in the equation to get the new y value, which is \(y_{new} = k \times \frac{x}{2}\).
5. Simplify the expression for the new y: This simplifies to \(y_{new} = \frac{1}{2} \times (k \times x)\), which shows that the new y is half of the original y.

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 Halving a Variable

Halving a variable means multiplying it by 1/2. In direct variation, if x is halved, y will also be multiplied by 1/2, since y changes proportionally to x.

Proportional Relationships and Constants

In proportional relationships, the constant of proportionality (k) remains fixed. Understanding that k does not change helps predict how y responds when x changes, such as when x is halved.