CMClearMathAcademy

Direct and Inverse Variation

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

Direct variation: y = kx, where k is the constant of variation. Doubling x doubles y. Inverse variation: y = k/x. Doubling x halves y. To find k from a single (x, y) pair, substitute and solve. In Functions, Linear Relationships, and Rate of Change, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.

What you'll learn

Why it matters: Direct: distance = rate * time (constant rate), cost = price-per-unit * quantity. Inverse: time to finish a job decreases as workers increase; pressure increases as volume decreases (Boyle's Law).

Worked example

Problem. y varies directly with x. When x = 4, y = 20. Find y when x = 7.

  1. Find k from the given pair: 20 = k(4), so k = 5.
  2. Use y = 5x with x = 7.
  3. y = 5(7) = 35.
  4. Connect the result back to Direct and Inverse Variation so the method and meaning are both clear.

Answer: 35

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Is y = 3x direct or inverse variation?

Choices: Direct · Inverse

Show solution
  1. y = kx form with k = 3.
  2. That is direct.

Answer: Direct

2. Is y = 6/x direct or inverse variation?

Choices: Direct · Inverse

Show solution
  1. y = k/x form with k = 6.
  2. That is inverse.

Answer: Inverse

3. If y varies directly with x and y = 12 when x = 4, find k.

Show solution
  1. Use y = kx: 12 = k(4).
  2. k = 3.

Answer: 3

4. If y varies inversely with x and y = 6 when x = 2, find k.

Show solution
  1. Use y = k/x: 6 = k/2.
  2. k = 12.

Answer: 12

5. y = 4x. Find y when x = 3.

Show solution
  1. y = 4(3) = 12.

Answer: 12

Practice this interactively with instant feedback and an AI tutor.

Practice Direct and Inverse Variation Take the free placement check

More Algebra I lessons