Education

Rational Equation Calculator

Solve rational equations while checking denominator restrictions and rejecting extraneous answers that do not work in the original equation.

rational-equation-calculator
Solve rational equations of the form (ax + b)/(cx + d) = (ex + f)/(gx + h). The calculator cross-multiplies, solves the resulting equation, and rejects values that make a denominator zero.
(ax + b)/(cx + d) = (ex + f)/(gx + h)
Solution
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Restrictions—
Equation after cross multiplication—
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How rational equations are solved

A rational equation is an equation that contains one or more rational expressions. The usual strategy is to clear the denominators, but the first step should always be to identify values that make any denominator zero. OpenStax covers the same denominator-aware thinking in its rational expressions section, where denominator restrictions are part of the expression itself.

After restrictions are identified, cross multiplication can turn a proportion into a polynomial equation. The final answer must be checked in the original equation because cross multiplication can produce an extraneous solution when a forbidden denominator value appears.

Worked example

(x + 2)/(x - 1) = (2x + 1)/(x + 3)\n\nRestrictions: x ≠ 1 and x ≠ -3\n\nCross multiply:\n(x + 2)(x + 3) = (2x + 1)(x - 1)\n\nx² + 5x + 6 = 2x² - x - 1\n0 = x² - 6x - 7\n0 = (x - 7)(x + 1)\n\nSolutions: x = 7 or x = -1

Neither solution violates the restrictions, so both are valid.

Why checking the answer matters

If a solution makes any original denominator zero, it is not a real solution even if it appears after algebraic manipulation. This is one of the most common places students lose marks: they solve the polynomial correctly but forget that the rational equation had domain restrictions from the beginning.

Common questions

  • A rational equation is an equation that contains at least one rational expression, meaning a fraction with a variable in the numerator or denominator. Examples include 1/(x - 2) = 3, (x + 1)/(x - 4) = 2, and (x + 2)/(x - 1) = (2x + 1)/(x + 3).
  • Start by finding excluded values from all denominators. Then clear denominators by multiplying by the least common denominator, or cross-multiply if the equation is one fraction equal to another fraction. Solve the resulting equation, then check each answer in the original rational equation.
  • An extraneous solution is a value that appears during the algebra but does not work in the original equation. In rational equations, this often happens when the value makes a denominator zero. That value must be rejected because division by zero is undefined.
  • Cross multiplication is valid for a proportion, meaning one fraction equals another fraction, as long as you still respect the original denominator restrictions. It is not a shortcut for every rational equation with several terms. For equations with multiple rational terms added or subtracted, use the least common denominator method instead.
  • The original denominators determine where the equation is defined. Even if a factor disappears later, the original equation was never allowed to use that x-value. That is why the calculator lists restrictions before solving and checks solutions at the end.