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!