Fractional Equations

Solve equations containing variables in the numerators or denominators. Learn to clear rational fractions using Least Common Denominators (LCD) and master cross-multiplication checks.

Algebra Concepts

What is a Fractional Equation?

A **fractional equation** (or rational equation) is an equation containing one or more rational expressions. The critical difference from linear equations is that the unknown variable can appear in the denominator.

Core Solving Strategies


  • Clear Denominators (LCD): Multiply both sides of the equation by the Least Common Denominator (LCD) of all rational terms to obtain a polynomial equation.
  • Cross-Multiplication (Butterfly Method): If the equation has the form A/B = C/D, multiply diagonally to get A · D = B · C.
  • Check for Extraneous Solutions: Any solved value that makes a denominator in the *original* equation zero (0) must be discarded as invalid.

Example Verification


6
x
+
4
x
= 5


Multiplying both sides by x clears the denominators:
x · (6/x + 4/x) = 5 · x ⇒ 6 + 4 = 5x
10 = 5x ⇒ x = 2

Since x = 2 does not make the original denominator x zero, it is a valid solution!