Equations with fractions

Why multiplying EVERY term by a common denominator clears the fractions, where cross-multiplication comes from (and its bracket discipline), and how one clearing move turns any fractional equation into an ordinary one.

Algebra · Linear equations

Equations with fractions

Why it works

Fractions in an equation are cleared, not fought. Multiply both sides by a common denominator and every fraction cancels away — because multiplying is done to the whole of each side, the balance survives.

\frac{x}{5} + 3 = 10

, multiplied through by 5:

x + 15 = 50 \;\Rightarrow\; x = 35.

Watch the 3: EVERY term gets multiplied, including the ones that weren't fractions. (

x + 3 = 50

is the classic wreck — the 3 escaped.)

A fraction each side: cross-multiplication, with brackets.

\frac{2x+1}{3} = \frac{x-2}{4}

— multiplying both sides by 3 and by 4 lands each denominator on the opposite numerator:

4(2x + 1) = 3(x - 2).

The brackets are not decoration: each multiplier hits the WHOLE numerator. Then expand and solve:

8x + 4 = 3x - 6

, so

5x = -10

x = -2

. Cross-multiplying is legal only in this one-fraction-each-side shape — an extra term breaks it (clear denominators instead).

Two fractions on one side: one multiplier clears both. Use the LCM of the denominators. For

\frac{x}{4} - \frac{x}{6} = 2

, multiply through by 12:

3x - 2x = 24 \;\Rightarrow\; x = 24.

Each numerator picks up only its share of the 12 (

12 \div 4 = 3

12 \div 6 = 2

) — and the right-hand side gets the full 12. The denominators themselves never add or subtract with each other.

Then it's just a linear equation. One clearing move is the whole trick; after it, the ordinary balance rules finish the job. Substitute back into the ORIGINAL fractional equation to check:

\frac{24}{4} - \frac{24}{6} = 6 - 4 = 2

. ✓