Guides / Equations

Solving equations with fractions: clear the denominators first

One multiplication turns a fraction equation into an ordinary one. Here's how to do it, with every step shown.

Why it works

You can multiply both sides of an equation by the same non-zero number without changing its solutions. If you choose a number every denominator divides into, each fraction turns into a whole-number expression. Working with 4x + 3x = 84 is much easier than working with twelfths.

Example 1: x/3 + x/4 = 7

The denominators are 3 and 4. The LCD is 12.

1. x3 + x4 = 7Start.
2. 12 · x3 + 12 · x4 = 12 · 7Multiply every term by 12, including the 7.
3. 4x + 3x = 8412 ÷ 3 = 4 and 12 ÷ 4 = 3.
4. 7x = 84Combine like terms.
5. x = 12Divide by 7.

Check: 12/3 + 12/4 = 4 + 3 = 7.

Example 2: numerators with more than one term

Solve x + 12 − x − 23 = 2. The LCD of 2 and 3 is 6.

1. x + 12 − x − 23 = 2Start.
2. 3(x + 1) − 2(x − 2) = 12Multiply every term by 6. Keep the numerators in parentheses.
3. 3x + 3 − 2x + 4 = 12Distribute. −2 times −2 is +4.
4. x + 7 = 12Combine like terms.
5. x = 5Subtract 7.

Check: (5 + 1)/2 − (5 − 2)/3 = 3 − 1 = 2.

Example 3: a variable in the denominator

Solve 3x + 12 = 54. The LCD is 4x. Because x is in a denominator, x can't be 0, so note that before you start.

1. 3x + 12 = 54Start. x ≠ 0.
2. 12 + 2x = 5xMultiply every term by 4x: 4x · 3/x = 12, 4x · 1/2 = 2x, 4x · 5/4 = 5x.
3. 12 = 3xSubtract 2x.
4. x = 4Divide by 3. It isn't 0, so it's allowed.

Check: 3/4 + 1/2 = 3/4 + 2/4 = 5/4.

Always check answers when the variable is in a denominator. Multiplying by an expression like x − 2 can produce an answer that makes a denominator zero. For example, clearing xx − 2 = 2x − 2 + 3 gives x = 2, which isn't allowed, so that equation has no solution.

Common mistakes, and the exact line they happen on

1. Forgetting to multiply the terms that aren't fractions

1. x3 + 2 = 5Start.
2. x + 2 = 15This is the mistake. The x/3 and the 5 were multiplied by 3, but the 2 wasn't.
3. x = 13Checking: 13/3 + 2 isn't 5.

1. x + 6 = 15Correct: 3 · 2 = 6.
2. x = 9Check: 9/3 + 2 = 5.

Multiplying “both sides” means multiplying each whole side, and each side here is a sum. Every term on it gets multiplied.

2. Losing a sign after a subtracted fraction

Back to Example 2. Without parentheses, the minus sign only reaches the first term of the second numerator:

1. 3(x + 1) − 2(x − 2) = 12Fine so far.
2. 3x + 3 − 2x − 4 = 12This is the mistake. −2 · (−2) is +4, not −4.
3. x = 13Checking: 14/2 − 11/3 isn't 2.

The fraction bar works like parentheses. When you clear it, write the parentheses in so the minus sign reaches every term.

3. Cross-multiplying when there are more than two fractions

Cross-multiplication is a shortcut for exactly one fraction on each side, like ab = cd. With three terms, as in Example 1, it doesn't apply. Multiplying by the LCD always works, including when there's one fraction on each side.

Practice

1. Solve x5 − 1 = 3.

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   Multiply by 5: x − 5 = 15, so x = 20.

2. Solve x2 + x3 = 10.

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   LCD 6: 3x + 2x = 60, so x = 12.

3. Solve 2x − 13 = x + 42.

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   LCD 6: 2(2x − 1) = 3(x + 4), 4x − 2 = 3x + 12, so x = 14. Check: 27/3 = 9 and 18/2 = 9.

4. Solve x − 34 − x + 16 = 1.

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   LCD 12: 3(x − 3) − 2(x + 1) = 12, 3x − 9 − 2x − 2 = 12, x − 11 = 12, so x = 23. Check: 20/4 − 24/6 = 5 − 4 = 1.

5. Solve 2x + 13 = 1.

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   LCD 3x (x ≠ 0): 6 + x = 3x, so x = 3. Check: 2/3 + 1/3 = 1.

6. Solve x4 + 12 = x3.

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   LCD 12: 3x + 6 = 4x, so x = 6. Check: 6/4 + 1/2 = 2 and 6/3 = 2.