Clear every denominator
First find the least common denominator, or LCD, of every fraction in the equation. Multiply both sides by that number. Because the same nonzero value multiplies both sides, the new equation preserves the original equality.
Treat each side as one complete expression and distribute the LCD to every term. Each denominator should divide evenly into the LCD, leaving an equivalent equation that is usually easier to solve.
Solve, then check the original
After clearing the denominators, combine like terms and isolate the variable using the same equation-solving strategies you already know. Simplifying each product before moving terms helps prevent sign and multiplication errors.
Substitute the result into the original fraction equation, not only the cleared version. This check can reveal a skipped term or an LCD that was not distributed across the entire equation. This lesson uses numerical denominators; equations with variables in denominators require additional restrictions.
Remember these ideas
Key ideas
- The LCD is the least common multiple of all numerical denominators in the equation.
- Multiply both complete sides by the LCD to preserve equality.
- Distribute the LCD to every term before simplifying.
- Check the solution in the original equation containing the fractions.
Make the reasoning visible
Worked example · x/3 + 1/2 = 5/6
- 1
The denominators are 3, 2, and 6, so the LCD is 6.
- 2
Multiply both sides by 6 and distribute: 6(x/3) + 6(1/2) = 6(5/6).
- 3
Simplify: 2x + 3 = 5. Subtract 3 to get 2x = 2, then divide by 2: x = 1.
- 4
Check in the original equation: 1/3 + 1/2 = 2/6 + 3/6 = 5/6, so the solution works.
Try the next step
Clear, solve, and check
Solve y/4 − 1/3 = 2/3. Begin by writing the equivalent equation produced when every term is multiplied by the LCD.
Why would multiplying only the term y/4—and not the other terms—change the equation?