## Solving a Rational Equation For a Specific Variable

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When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.

We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.

## Example

Solve: \(\frac{D}{T}=R\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}T.\)

### Solution

Note any value of the variable that would make any denominator zero. | |

Clear the fractions by multiplying both sides ofthe equations by the LCD, T. | |

Simplify. | |

Divide both sides by R to isolate T. | |

Simplify. |

The example below uses the formula for slope that we used to get the point-slope form of an equation of a line.

## Example

Solve: \(m=\frac{x-2}{y-3}\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}y.\)

### Solution

Note any value of the variable that would make any denominator zero. | |

Clear the fractions by multiplying both sides ofthe equations by the LCD, \(y-3\). | |

Simplify. | |

Isolate the term with y. | |

Divide both sides by m to isolate y. | |

Simplify. |

Be sure to follow all the steps in the example below. It may look like a very simple formula, but we cannot solve it instantly for either denominator.

## Example

Solve \(\frac{1}{c}+\frac{1}{m}=1\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}c.\)

### Solution

Note any value of the variable that would make any denominator zero. | |

Clear the fractions by multiplying both sides ofthe equations by the LCD, \(cm\). | |

Distribute. | |

Simplify. | |

Collect the terms with c to the right. | |

Factor the expression on the right. | |

To isolate c, divide both sides by \(m-1\). | |

Simplify by removing common factors. |

Notice that even though we excluded \(c=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}m=0\) from the original equation, we must also now state that \(m\ne 1\).