Clearing Denominators

Algebra 1 · Axiom Academy

Multiply every term by the LCD so all denominators cancel — turning a fraction equation into a clean polynomial you can just solve. 1. The Move: Wipe Out Every Denominator Fractions make an equation tedious. The strategy of clearing denominators multiplies both sides by the LCD , so each denominator cancels and the equation becomes one with no fractions at all — far easier to solve. The fraction equation (hard to handle) After multiplying every term by the LCD = 12 2. Building the LCD from Prime Factors Before clearing, find the least common denominator — the smallest number every denominator divides into evenly. The reliable way: factor each denominator into primes, then take each prime the most times it appears in any one of them. Take two 2 's (the most, from 4 ) and one 3 (from 6 or 3 ): With the LCD in hand, multiply every term on both sides by it. Inside each term the LCD and the denominator reduce — 12 over 4 leaves 3 , 12 over 6 leaves 2 , 12 over 3 leaves 4 — and every fraction collapses to a whole-number coefficient. 4. The Catch: Extraneous Solutions Here is the critical catch. When the LCD itself contains the variable, multiplying by it can introduce an extraneous solution — a value that satisfies the cleared equation but makes a denominator zero in the original . The algebra hands it to you; the original equation forbids it. An extraneous solution in action Division by zero — x = 2 is extraneous and must be rejected. This equation has no valid solution.

This is the written version of the interactive lesson above. See the full Algebra 1 course.