Linear Equations • Topic 3 of 4

Equations with Fractions

Fractions inside an equation are best cleared immediately by multiplying every term — on both sides — by the least common denominator. That single move turns a fraction-heavy equation into a clean integer one and eliminates the arithmetic that trips people up under time pressure. To solve x/3 + x/4 = 7, multiply through by 12: 4x + 3x = 84, so 7x = 84 and x = 12. Multiply every term, including any whole-number terms, not just the fractions. When the variable appears in a denominator you can cross-multiply, but then check the answer does not make an original denominator zero. Proportions — one fraction equal to another — are the special case where cross-multiplication is fastest: a/b = c/d becomes ad = bc.

✅ Solved examples

1. Solve x/2 + x/3 = 5.
LCD is 6. Multiply through: 3x + 2x = 30, so 5x = 30, x = 6.
2. Solve (x + 1)/4 = 3.
Multiply both sides by 4: x + 1 = 12, so x = 11.
3. Solve (2x)/5 − 1 = 3.
Add 1: (2x)/5 = 4. Multiply by 5: 2x = 20, so x = 10.
4. Solve the proportion 3/x = 9/12.
Cross-multiply: 3·12 = 9x, so 36 = 9x and x = 4.

✏️ Practice — try these, take hints as needed

1. Solve x/3 + x/6 = 4.
LCD is 6.
Multiply through: 2x + x = 24.
3x = 24.
x = 8
2. Solve (x − 2)/5 = 4.
Multiply by 5.
x − 2 = 20.
Add 2.
x = 22
3. Solve (3x)/4 + 2 = 8.
Subtract 2.
(3x)/4 = 6.
Multiply by 4, then divide by 3.
x = 8
4. Solve the proportion x/6 = 10/15.
Cross-multiply.
15x = 60.
Divide by 15.
x = 4
5. Solve x/2 − x/5 = 3.
LCD is 10.
5x − 2x = 30.
3x = 30.
x = 10

📝 Topic test — 8 questions

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