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Direct, Inverse, Joint, and Combined Variation

A free College Algebra lesson from the “Systems, Matrices, and Linear Programming” unit, with a worked example and practice problems including step-by-step solutions.

DIRECT: y = kx. INVERSE: y = k/x. JOINT: y varies jointly as x and z means y = kxz. COMBINED: mixes direct and inverse, e.g., y = kxz / w. In every case, find k from a known data point and then use the equation to find unknowns.

What you'll learn

Why it matters: Boyle's Law (P varies inversely with V), the area of a triangle (jointly with base and height), and gravitational force (combined: directly with masses, inversely with distance squared) are all variation models.

Worked example

Problem. y varies jointly with x and z. When x = 2 and z = 3, y = 24. Find y when x = 5 and z = 4.

  1. Joint: y = k * x * z. Find k: 24 = k * 2 * 3 = 6k, so k = 4.
  2. Then y = 4 * 5 * 4 = 80.

Answer: 80

Practice problems

1. y varies directly with x; y = 12 when x = 4. Find k.

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

Answer: 3

2. y varies inversely with x; y = 6 when x = 2. Find k.

Show solution
  1. Warm-up: First identify exactly what the question is asking: y varies inversely with x; 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. k = 6 * 2 = 12.
  4. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

3. y varies jointly with x and z; y = 24 when x = 2, z = 3. Find k.

Show solution
  1. Warm-up: First identify exactly what the question is asking: y varies jointly with x and z; y = 24 when x = 2, z = 3. Find k.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. k = y / (xz) = 24 / 6 = 4.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. y = 4xz. Find y when x = 5 and z = 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: y = 4xz. Find y when x = 5 and z = 4.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. y = 4 * 5 * 4 = 80.
  4. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

5. y varies directly with x and inversely with z. y = 10 when x = 4, z = 2. Find k.

Show solution
  1. Core Practice: First identify exactly what the question is asking: y varies directly with x and inversely with z. y = 10 when x = 4, z = 2. Find k.
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. y = kx/z, so 10 = k * 4 / 2 = 2k, k = 5.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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