Equations with fractions

1Summary 重點摘要

For equations with fractions: multiply every term on both sides by the LCM to clear denominators, then solve as usual.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a fractional equation?A fractional equation is one where the unknown appears in a fraction, such as x2 = 5 with x in the numerator; x3 + 1 = 4 is another. Their special feature is that the denominator obstructs direct solving, so the first technique is to clear the fractions and turn the equation into an ordinary one. A daily example: half of a rope measures 5 m — how long was it? Let the original length be x m, giving x2 = 5. Once the clearing technique is mastered, the rest is identical to any linear equation.
2. 2. Step 1: Clear fractionStep one is clearing the fraction: multiply both sides by the denominator. For x2 = 5 the denominator is 2, so multiply both sides by 2: 2 × x2 = 5 × 2; the 2 and the denominator cancel on the left, giving x = 10. The balance principle applies — the same operation on both sides changes nothing. It must be both sides, and later, every term on both sides; multiplying one side only breaks the balance, which is the fundamental mistake to avoid.
3. 3. Step 2: SolveStep two handles an equation with both a fraction and a constant. In x3 + 1 = 4 there are two parts: the fraction term and the constant term. Clear the fraction first: multiply every term on both sides by 3 — 3 × x3 + 3 × 1 = 3 × 4 — noting that the constant 1 is multiplied too, giving x + 3 = 12. The classic error is exactly forgetting the constant, writing x + 1 = 12 and ruining everything; remember 'multiply every term' and most fractional-equation errors disappear.
4. 4. Step 3: TransposeStep three is transposing to finish. From x + 3 = 12, move +3 across as −3: x = 12 − 3 = 9. The rule matches earlier lessons — a term flips sign when it crosses the equals sign. Review the whole flow: multiply by the denominator to clear fractions, transpose to isolate x, then compute. Each stage feeds the next, so a sign error anywhere reaches the end; check the signs after each step rather than once at the finish.
5. 5. CheckChecking substitutes the solution into the original equation: with x = 9, x3 + 1 = 93 + 1 = 3 + 1 = 4, matching the right side, so the answer is correct. The check is especially valuable here because clearing fractions involves several steps where one missed term or wrong cancellation hides easily. If x = 6 had been obtained, substitution gives 63 + 1 = 3, not 4, exposing the error at once. Make 'substitute after solving' a fixed habit and accuracy rises noticeably.
6. 6. Two fractionsFor two fractions, as in x2 + x3 = 5, first find the lowest common denominator: the LCM of 2 and 3 is 6. Then multiply every term on both sides by 6: 6 × x2 + 6 × x3 = 6 × 5, which after cancelling gives 3x + 2x = 30. All three terms are multiplied, including the 5 on the right; multiplying only the left two terms is the common error. One common denominator clears all fractions at once — far faster than handling them one by one — and is the core technique of the topic.
7. 7. SimplifyAfter clearing, the equation is an ordinary linear one. In 3x + 2x = 30, combine like terms: 3x + 2x = 5x, so 5x = 30; divide both sides by 5 to get x = 6. Finally check: 62 + 63 = 3 + 2 = 5, matching the original right side. The full flow is: find the LCM, multiply every term on both sides, combine like terms, transpose and solve, substitute to verify. Step two — the missed term — remains the likeliest mistake, so point through the terms one by one after multiplying.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Fractional: variable in fraction
• Step 1: Clear fractions (multiply denominator)
• Two fractions: find LCD
• After clearing, it's normal equation
• Check: substitute back