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How to Solve Equations With Fractions

Clear the fractions first, then solve like a regular equation. One simple trick makes these problems easy.

📐 Algebra⏱️ ~12 min read📊 Intermediate

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What You'll Learn

  • The "clear the fractions" method (LCD technique)
  • How to find the LCD quickly
  • Handling equations with multiple fractions
  • When variables appear in denominators

The Key Insight

Fractions make equations look scary, but there's a simple fix: multiply every term by the LCD (least common denominator) to eliminate all fractions at once. Then you have a regular equation.

💡 The Golden Rule

Whatever you multiply one side by, you must multiply the OTHER side by too—including every term.

For more on working with fractions in algebra, see Math is Fun's guide.

The LCD Method (Step by Step)

Step 1: Find the LCD

Look at all the denominators in your equation. Find their least common multiple. This is your LCD.

Example: If denominators are 2, 3, and 6

LCD = 6 (since 6 is divisible by 2, 3, and 6)

Step 2: Multiply Every Term by the LCD

Multiply every single term on both sides by the LCD. Don't forget terms without fractions!

Step 3: Simplify (Fractions Disappear!)

The LCD cancels with each denominator, leaving you with a fraction-free equation.

Step 4: Solve the Equation

Now solve using standard methods (combine like terms, isolate x).

Step 5: Check Your Answer

Substitute back into the original equation (with fractions) to verify.

Worked Examples

Example 1: Simple fraction equation

Solve: x/4 + 3 = 7

x/4 + 3 = 7
LCD = 4. Multiply every term by 4:
4·(x/4) + 4·(3) = 4·(7)
x + 12 = 28
x = 28 - 12
x = 16
Check: 16/4 + 3 = 4 + 3 = 7 ✓

Example 2: Multiple fractions

Solve: x/2 + x/3 = 5

x/2 + x/3 = 5
LCD = 6. Multiply every term by 6:
6·(x/2) + 6·(x/3) = 6·(5)
3x + 2x = 30
5x = 30
x = 6
Check: 6/2 + 6/3 = 3 + 2 = 5 ✓

Example 3: Fractions on both sides

Solve: (2x - 1)/3 = (x + 2)/4

(2x - 1)/3 = (x + 2)/4
LCD = 12. Multiply both sides by 12:
12·(2x - 1)/3 = 12·(x + 2)/4
4(2x - 1) = 3(x + 2)
8x - 4 = 3x + 6
8x - 3x = 6 + 4
5x = 10
x = 2

Example 4: Variable in denominator (careful!)

Solve: 5/x = 2

5/x = 2
Multiply both sides by x:
5 = 2x
x = 5/2 = 2.5
Check: 5/(5/2) = 5 × (2/5) = 2 ✓
⚠️ Important: x ≠ 0 (can't divide by zero)

Quick LCD Reference

DenominatorsLCD
2, 44
2, 36
3, 412
2, 510
4, 612
2, 3, 412

Common Mistakes to Avoid

⚠️

Forgetting to multiply ALL terms

If you multiply x/2 by 6, you must also multiply the other side by 6. Every term!

⚠️

Wrong LCD

For denominators 4 and 6, the LCD is 12 (not 24). Use the LEAST common multiple.

⚠️

Forgetting to distribute after clearing

After 12·(2x-1)/3 = 4(2x-1), remember: 4(2x-1) = 8x - 4, not 8x - 1.

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How to Solve Equations With Fractions | Step-by-Step Guide