If x and y are variables and k is a constant, which of the following describes an inverse relationship between x and y?

Analyze the question: We need the equation that shows an inverse (indirect) variation between x and y.

Concept:

• Direct variation: y ∝ x, so y = kx. When x increases, y increases proportionally.
• Inverse variation: y ∝ 1/x, so y = k/x, or equivalently xy = k. When x increases, y decreases, and the product xy stays constant.

Checking each option:

• A. y = kx is direct variation (y/x = k is constant).
• B. y = k/x is inverse variation, because xy = k is constant.
• C. y = k√x is a direct variation with the square root of x (y ∝ √x), not an inverse one.
• D. y = x + k is a linear relationship with an added constant. It is not a variation at all, because y does not change proportionally with x.

Quick check with numbers (k = 12): If x = 2, y = 6. If x = 4, y = 3. When x doubles, y halves, and xy = 12 both times. This is the signature of inverse variation.

Answer: B

Exam tip: Look for the phrase "inversely proportional" or "varies inversely". Write it as y = k/x, then use a given pair of values to find k. Remember: direct = same direction (y = kx), inverse = opposite direction (xy = k).