Linear Equations with Fractions

Article Contents

1. Equations with Fractions
2. Unknown in the Numerator
3. Unknown in the Denominator
4. Clearing Fractions Early
5. Examples

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

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

Example with Fractions

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

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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

$$\frac{28}{4}=7✓$$

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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 $x$

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

Step 2: Divide both sides by 2

$$x=3$$

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

$$\frac{6}{3}=2✓$$

Important: $x\ne 0$ since we can't divide by zero!

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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.

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5. Examples {#examples}

Example 1: Unknown in Numerator

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

```

Step 1: Multiply both sides by 3

2x = 24

Step 2: Divide both sides by 2

x = 12

```

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

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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: $\frac{12}{3}=4✓$

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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: $\frac{9+5}{2}=\frac{14}{2}=7✓$

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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: $x=a/b$
$\frac{ax}{b}=c$	Multiply by $b$, then divide by $a$: $x=bc/a$

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Interactive Exercises

• Fractions with Unknown in Numerator
• Fractions with Unknown in Denominator
• Both Sides with Fractions

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Related Articles

• Comprehensive Guide - Full techniques
• Simple Equations - Basic equations