Overview
Multiplying every term of an equation by a common denominator turns fractional coefficients into integers without changing the solution. Term boundaries matter: a numerator inside a fraction is an entire group. Decimals can be treated as exact base-ten fractions.
Consider this example: Solve x/3 + 2 = 5. Subtract 2 to get x/3 = 3; multiply by 3 to obtain x = 9. The choice 3 is the value of x/3, not x.
The method also handles a different case: Solve (x − 1)/4 + (x + 2)/3 = 3. Multiply every term by 12: 3(x − 1) + 4(x + 2) = 36. This becomes 7x + 5 = 36, so x = 31/7.
For a further application, consider: Solve 3 − (x + 2)/2 = (x − 4)/3. Multiply by 6: 18 − 3(x + 2) = 2(x − 4). Expanding gives 12 − 3x = 2x − 8, so 20 = 5x and x = 4. The outside subtraction applies to both terms of x + 2.
Multiplying only the fractions and leaving a whole-number term unchanged does not perform the same operation on both sides. Choose a common denominator, apply it to every term, and check the answer in the original equation. For decimals, multiplying by 10 or 100 often makes the structure clearer.