Use a Constant Product for Inverse Variation

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A positive inverse-variation rule y=k/x passes through (2,3). Find y at x=4. Candidate answers: (A) 0.5 (B) 2.5 (C) 3 (D) 3/2.
• Independent check inputs: A positive inverse-variation rule y=k/x passes through (3,4). Find y at x=6. Candidate answers: (A) 1 (B) 2 (C) 3 (D) 4.

Learn the idea

Rational expressions divide polynomials, so denominator zeros are excluded from their domains. Factoring can reveal cancellation or help solve equations, but it does not erase the original restrictions or turn inverse variation into linear change. Inverse variation has the form y=k/x. A supplied nonzero pair determines the constant product xy=k, and a new input recovers its output by division.

Here is the method on the supplied teaching case. First k=xy=2×3=6. Then y=6/4=3/2. Doubling x halves y; inverse variation is not a constant-difference rule.

Inverse variation maintains a constant product rather than a constant difference or ratio y/x. Zero input is excluded; doubling a positive input halves its output under the supplied positive model.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A positive inverse-variation rule y=k/x passes through (9,10). Find y at x=18. Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6. Show the mathematical reason, not only the selected answer. Expected result: 5. First k=xy=9×10=90. Then y=90/18=5. Doubling x halves y; inverse variation is not a constant-difference rule.

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