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Solving a Rational Equation for a Specific Variable
When a formula takes the form of a rational equation — meaning the variable to be isolated appears in a denominator — the same five-step strategy used for solving rational equations must be combined with the technique for solving a formula for a specific variable. Many formulas in business, science, economics, and other fields are rational equations that relate two or more variables, so this skill arises frequently.
The procedure follows the same steps used for any rational equation:
- Note any value of the variable that would make a denominator zero. Even though the goal is to express one variable in terms of others rather than find a numerical answer, restricted values must still be identified.
- Clear the fractions by multiplying both sides of the equation by the LCD.
- Simplify the resulting equation.
- Isolate the target variable using standard equation-solving techniques — distributing, collecting all terms containing the target variable on one side, factoring the target variable out if it appears in more than one term, and dividing.
- Check that the result does not violate any of the restrictions identified in Step 1.
The key difference from earlier formula-solving problems is that the target variable now appears in a denominator, so multiplying by the LCD is necessary before the variable can be isolated. The final answer is an expression in terms of the remaining variables rather than a single number.
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Ch.8 Rational Expressions and Equations - Elementary Algebra @ OpenStax
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