AlgebraStructure & Method · Book 1All Chapters

Variation

12-8Joint and Combined Variation

Overview

Joint: y = kxz (varies jointly). Combined: y = kx/z (direct with one, inverse with another).

Key ideas

  • Combine direct and inverse relationships.
  • Find k from one set of values.
  • Substitute to predict.

Worked examples

Example 1

Problem. z varies jointly as x and y. z = 30 when x = 2, y = 5. Find z when x = 3, y = 4.

Solution. k = 3. z = 3·3·4 = 36.

Practice

Try each one. Click Show answer when ready.

  1. 1.

    z = kxy, z = 24 at x = 2, y = 3. Find k.

  2. 2.

    y = kx/z; y = 6 at x = 4, z = 2. Find y at x = 8, z = 4.

  3. 3.

    z = kxy; k = 2, x = 5, y = 3. Find z.

Challenge problems

A little tougher — great for test prep. Click Show answer when ready.

  1. 1.

    z varies jointly with x and y. If z = 24 when x = 2 and y = 3, find z when x = 4 and y = 5.

  2. 2.

    y varies directly with x and inversely with z². If y = 6 when x = 4 and z = 2, find y when x = 9 and z = 3.