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Combined Variation

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Combined Variation

Study Guide

Key Definition

Combined variation describes a situation where a variable depends on two or more other variables, and varies directly with some of them and varies inversely with others.

Important Notes

A situation can involve both direct and inverse variation.
A single constant of variation (k) governs the entire combined relationship, not separate constants for direct and inverse parts.
Remember to isolate the variable when solving combined variation problems.

Mathematical Notation

$z = k \cdot \frac{x}{y}$ means z varies directly as x and inversely as y.

$x = k \cdot \frac{y}{z}$ means x varies directly as y and inversely as z.

Remember to use proper notation when solving problems

Why It Works

Combined variation works because it allows us to model situations where the relationship between variables is complex and depends on more than one factor.

Remember

A single constant k governs the entire combined relationship.

Quick Reference

Direct Variation:$y = kx$

Inverse Variation:$y = \frac{k}{x}$

Understanding Combined Variation

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Video explanation of this concept

Beginner

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Beginner Explanation

Combined variation is when a variable depends and varies directly with some variables and inversely with others.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If z varies directly as x and inversely as y, and z = 6 when x = 3 and y = 2, what is the value of z when x = 4 and y = 3?

Real-World Problem

Question Exercise

Intermediate

Physics Scenario

In physics, the intensity of sound I varies directly as the square of the amplitude A and inversely as the square of the distance D. If I = 10 when A = 5 and D = 3, what is the intensity of sound when A = 4 and D = 2?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If x varies directly as y and inversely as the cube of z, and x = 10 when y = 5 and z = 3, for what value of z will x = 3 and y = 4?

Challenge Quiz

Single Choice Quiz

Advanced

If x varies directly as the square of y and inversely as the cube root of z, and x = 10 when y = 2 and z = 8, what is the value of x when y = 3 and z = 27?

Recap

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