Lesson 3.10
Solving via LCD
When a rational equation has three or more terms, cross-multiplication won't work. Instead, multiply the entire equation by the LCD to clear all denominators at once.
Introduction
Cross-multiplication only works when there's exactly one fraction on each side. For equations like , we need a more powerful tool: multiplying every term by the LCD.
Past Knowledge
Finding the LCD (Lesson 3.5) and solving proportions (Lesson 3.9).
Today's Goal
Multiply every term by the LCD to clear all fractions, then solve the resulting equation.
Future Success
This is the universal method for all rational equations. Lesson 3.11 covers what happens when solutions are extraneous.
Key Concepts
The LCD Clearing Method
Find the LCD of ALL denominators in the equation
Multiply EVERY term on BOTH sides by the LCD
Each denominator cancels, leaving a polynomial equation
Solve the resulting equation
Check answers against domain restrictions
Why Does This Work?
When you multiply a fraction by its own denominator, the denominator cancels:
The LCD contains every denominator as a factor, so it clears all fractions simultaneously.
Worked Examples
Example 1: Numeric Denominators
BasicSolve .
LCD = 6. Multiply every term by 6.
Simplify
✓
Example 2: Variable Denominators
IntermediateSolve .
LCD = . Multiply every term.
Simplify and solve
✓ (not zero)
Example 3: Factored Denominators
AdvancedSolve .
Factor: → LCD =
Multiply every term by LCD
Distribute and solve
✓ (not ±1)
Common Pitfalls
Multiplying Only Some Terms
You must multiply every single term on both sides by the LCD, including terms that aren't fractions (like the "5" in Example 1).
Distributing the Negative
In Example 3, , not . Parentheses matter!
Real-Life Applications
Work-rate problems are classic LCD applications: "If Pump A fills a pool in 4 hours and Pump B fills it in 6 hours, how long do they take together?" The equation is solved by multiplying through by the LCD.
Practice Quiz
Loading...