Linear Equations with Fractions

Article Contents

Equations with Fractions
Unknown in the Numerator
Unknown in the Denominator
Clearing Fractions Early
Examples

1. Equations with Fractions {#equations-with-fractions}

Fractions in equations can look scary, but the same balance principle applies.

Example with Fractions

\frac{x}{3} + 5 = 12

2. Unknown in the Numerator {#unknown-in-the-numerator}

This is the standard case: $\frac{x}{a} = b$

Formula: Multiply both sides by

a

Example

Solve: $\frac{x}{4} = 7$

Step 1: Multiply both sides by 4

\frac{x}{4} \times 4 = 7 \times 4

Step 2: Calculate

x = 28

Step 3: Check

\frac{28}{4} = 7 ✓

3. Unknown in the Denominator {#unknown-in-the-denominator}

When $x$ is in the denominator, we need to be careful.

Example

Solve: $\frac{6}{x} = 2$

Step 1: Multiply both sides by

x

\frac{6}{x} \times x = 2 \times x

6 = 2 x

Step 2: Divide both sides by 2

x = 3

Step 3: Check (remember: never divide by zero!)

\frac{6}{3} = 2 ✓

Important:

x \neq = 0

since we can't divide by zero!

4. Clearing Fractions Early {#clearing-fractions-early}

A helpful technique is to multiply the entire equation by the denominator to avoid working with fractions.

Example

Solve: $\frac{x + 2}{3} = 5$

Method 1: Direct approach

\frac{x + 2}{3} = 5 \Rightarrow x + 2 = 15 \Rightarrow x = 13

Method 2: Clear fractions first

\frac{x + 2}{3} = 5 \Rightarrow (x + 2) = 5 \times 3 = 15 \Rightarrow x = 13

Both methods work! Method 2 is often cleaner.

5. Examples {#examples}

Example 1: Unknown in Numerator

Solve: $\frac{2 x}{3} = 8$

Step 1: Multiply both sides by 3
        2x = 24
Step 2: Divide both sides by 2
        x = 12

Check: $\frac{2 \times 12}{3} = \frac{24}{3} = 8 ✓$

Example 2: Unknown in Denominator

Solve: $\frac{12}{x} = 4$

Step 1: Multiply both sides by x
        12 = 4x
Step 2: Divide both sides by 4
        x = 3

Check: $\frac{12}{3} = 4 ✓$

Example 3: Complex Fractions

Solve: $\frac{x + 5}{2} = 7$

Step 1: Multiply both sides by 2
        x + 5 = 14
Step 2: Subtract 5 from both sides
        x = 9

Check: $\frac{9 + 5}{2} = \frac{14}{2} = 7 ✓$

Summary

Situation	Method
$\frac{x}{a} = b$	Multiply both sides by $a$ : $x = ab$
$\frac{a}{x} = b$	Multiply both sides by $x$ , then divide: $x = a / b$
$\frac{a x}{b} = c$	Multiply by $b$ , then divide by $a$ : $x = b c / a$

Interactive Exercises

Comprehensive Guide - Full techniques
Simple Equations - Basic equations

Linear Equations with Fractions

Article Contents

1. Equations with Fractions {#equations-with-fractions}

Example with Fractions

2. Unknown in the Numerator {#unknown-in-the-numerator}

Example

3. Unknown in the Denominator {#unknown-in-the-denominator}

Example

4. Clearing Fractions Early {#clearing-fractions-early}

Example

5. Examples {#examples}

Example 1: Unknown in Numerator

Example 2: Unknown in Denominator

Example 3: Complex Fractions

Summary

Interactive Exercises

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