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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.

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.

Answer: 35

Practice problems

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

Choices: Direct · Inverse

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = 3x direct or inverse variation?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. y = kx form with k = 3.
  4. That is direct.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Direct

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

Choices: Direct · Inverse

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = 6/x direct or inverse variation?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. y = k/x form with k = 6.
  4. That is inverse.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Inverse

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: If y varies directly with x and y = 12 when x = 4, find k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use y = kx: 12 = k(4).
  4. k = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: If y varies inversely with x and y = 6 when x = 2, find k.
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Use y = k/x: 6 = k/2.
  4. k = 12.
  5. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: y = 4x. Find y when x = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. y = 4(3) = 12.
  4. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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