Lesson 2.3

One-Step Equations with Fractions

Fractions stop being scary when you learn their secret weapon: the reciprocal. One multiplication step is all it takes to clear any fraction.

Introduction

In Lesson 2.2, we divided to undo multiplication. But dividing by a fraction (like ) is messy. Instead, we multiply by the reciprocal. It's cleaner, faster, and less prone to errors.

Past Knowledge

You know how to multiply fractions: .

Today's Goal

Solve equations with fractional coefficients using reciprocals.

Future Success

This technique is essential for clearing denominators in complex rational equations later.

Key Concepts

1. The Reciprocal Rule

Multiplying a fraction by its reciprocal always equals 1. This leaves us with just , or .

2. Apply to Both Sides

To keep the balance, if you multiply the left side by , you must multiply the right side by .

3. Negative Fractions

The reciprocal has the same sign as the original number. The reciprocal of is , not .

Worked Examples

Example 1: Basic Fraction

Basic

Solve .

1

Find the Reciprocal

Flip to get .

2

Multiply Both Sides

3

Simplify

Example 2: Negative Fraction

Intermediate

Solve .

1

Multiply by Reciprocal

Reciprocal of is .

2

Simplify

Example 3: Fractional Solution

Advanced

Solve .

1

Multiply by Reciprocal

2

Calculate Right Side

Simplify by dividing by 2:

Common Pitfalls

Flipping the Sign

The reciprocal of is . Students often flip it to , which is wrong. Reciprocals keep the sign.

Dividing Fractions Improperly

Attempting to divide in your head often leads to errors. Always write out the multiplication step: .

Real-Life Applications

Gear ratios in cars and bicycles are fractions. If a gear reduces speed by , to get back to the original speed, you need a gear ratio of . Engineering is full of these reciprocal relationships.

Practice Quiz

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