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Inverse Variation

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Inverse Variation

Study Guide

Key Definition

In inverse variation, as one variable increases, the other decreases, represented by $x y = k$ or $y = \frac{k}{x}$.

Important Notes

In inverse variation, $x y = k$ is constant.
If $(x_1, y_1)$ and $(x_2, y_2)$ are solutions, then $x_1 y_1 = x_2 y_2$.
Graph of inverse variation is a hyperbola.
Inverse variation is not linear.
Product of variables remains constant.

Mathematical Notation

$\times$ represents multiplication

$/$ represents division

$x y = k$ equation for inverse variation

$k$ is a constant

$x$, $y$ are variables

Remember to use proper notation when solving problems

Why It Works

In inverse variation, the product $x y = k$ remains constant, ensuring that an increase in one variable results in a decrease in the other.

Remember

The key formula for inverse variation is $x y = k$.

Quick Reference

Inverse Variation Equation:$x y = k$

Alternate Form:$y = \frac{k}{x}$

Understanding Inverse Variation

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In inverse variation, the product $x y$ remains constant. As x increases, y decreases to maintain $xy = k$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $x y = 12$ and $x = 4$, what is $y$?

Real-World Problem

Question Exercise

Intermediate

Speed-Time Relationship

A car travels a distance where speed and time vary inversely. If the speed is $60$ km/h, the journey takes $2$ hours. How long will it take at $120$ km/h?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Suppose $x y = 25$ and $x$ doubles. What happens to $y$?

Challenge Quiz

Single Choice Quiz

Advanced

If $y = \frac{36}{x}$ and x increases by $50\%$, how does the new y compare to the original y?

Recap

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Review key concepts and takeaways