53.4 Linear Rational Equations

Linear Rational Equations are rational equations that, once every denominator has been cleared using the equation-wide LCD, reduce to a linear polynomial equation, meaning the variable appears with no exponent higher than one throughout the cleared equation. This is the simpler of the two outcome types a rational equation can produce after clearing, since the resulting linear equation can be solved directly using the standard technique of isolating the variable, without any need for factoring.

Even though the algebra involved in solving the cleared equation is straightforward, a linear rational equation still requires the same domain screening applied to every rational equation, since the clearing step can introduce a candidate solution that must ultimately be rejected.

Recognizing the Outcome

Cleared Linear Outcome

After the LCD has been distributed across every term of the original rational equation, the resulting equation is examined to confirm that the variable appears only to the first power throughout, with no squared or higher-degree term present, classifying it as a linear rational equation.

Simplifying the Cleared Equation

Rational Linear Term Consolidation

Every term of the cleared equation containing the variable is combined into a single term, using ordinary like-term combination, across both sides of the equation if necessary.

Rational Linear Constant Consolidation

Every constant term of the cleared equation is likewise combined into a single constant, keeping the variable terms and constant terms as two separate, fully consolidated groups.

Solving for the Candidate

Linear Candidate Isolation

The variable is isolated on one side of the consolidated equation using the standard linear equation techniques, addition or subtraction to move constant terms, and division to remove any remaining coefficient, producing a single candidate solution.

Applying the Required Screening

Linear Candidate Domain Screening

The single candidate solution obtained from isolating the variable is compared against the domain restriction determined during equation restriction preparation, checking whether it matches any of the originally excluded values.

Original Rational Equation Check

As a further check, the candidate is substituted directly into the original, uncleared rational equation to confirm it produces a true statement; this substitution both confirms the algebra was correct and independently catches any candidate that would make a denominator zero.

Reaching the Final Answer

Accepted Linear Solution

If the candidate passes both the domain screening and the substitution check, it is accepted as the valid solution to the linear rational equation; if the candidate fails either check, it is rejected, and the equation is reported as having no solution, since a linear equation produces only one candidate to begin with.