## Solving Equations with Fraction Coefficients

Contents

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\frac{1}{8}\phantom{\rule{0.1em}{0ex}}x+\frac{1}{2}=\frac{1}{4}.\)

To isolate the \(x\) term, subtract \(\frac{1}{2}\) from both sides. | |

Simplify the left side. | |

Change the constants to equivalent fractions with the LCD. | |

Subtract. | |

Multiply both sides by the reciprocal of \(\frac{1}{8}\). | |

Simplify. |

This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the **Multiplication Property of Equality** and multiply both sides of an equation by the least common denominator of *all* the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called *clearing the equation of fractions*. Let’s solve the same equation again, but this time use the method that clears the fractions.

## Example

Solve: \(\frac{1}{8}\phantom{\rule{0.1em}{0ex}}x+\frac{1}{2}=\frac{1}{4}.\)

### Solution

Find the least common denominator of all the fractions in the equation. | |

Multiply both sides of the equation by that LCD, 8. This clears the fractions. | |

Use the Distributive Property. | |

Simplify — and notice, no more fractions! | |

Solve using the General Strategy for Solving Linear Equations. | |

Simplify. | |

Check: Let \(x=-2\) |

Notice in the example above that once we cleared the equation of fractions, the equation was like those we solved earlier in this tutorial. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

### How to Solve equations with fraction coefficients by clearing the fractions.

- Find the least common denominator of
*all*the fractions in the equation. - Multiply both sides of the equation by that LCD. This clears the fractions.
- Solve using the General Strategy for Solving Linear Equations.

## Example

Solve: \(7=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x+\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x-\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x.\)

### Solution

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the least common denominator of all the fractions in the equation. | |

Multiply both sides of the equation by 12. | |

Distribute. | |

Simplify — and notice, no more fractions! | |

Combine like terms. | |

Divide by 7. | |

Simplify. | |

Check: Let \(x=12.\) | |

In the next example, we’ll have variables and fractions on both sides of the equation.

## Example

Solve: \(x+\frac{1}{3}=\frac{1}{6}\phantom{\rule{0.1em}{0ex}}x-\frac{1}{2}.\)

### Solution

Find the LCD of all the fractions in the equation. | |

Multiply both sides by the LCD. | |

Distribute. | |

Simplify — no more fractions! | |

Subtract \(x\) from both sides. | |

Simplify. | |

Subtract 2 from both sides. | |

Simplify. | |

Divide by 5. | |

Simplify. | |

Check: Substitute \(x=-1.\) | |

In the example below, we’ll start by using the Distributive Property. This step will clear the fractions right away!

## Example

Solve: \(1=\frac{1}{2}\left(4x+2\right).\)

### Solution

Distribute. | |

Simplify. Now there are no fractions to clear! | |

Subtract 1 from both sides. | |

Simplify. | |

Divide by 2. | |

Simplify. | |

Check: Let \(x=0.\) | |

Many times, there will still be fractions, even after distributing.

## Example

Solve: \(\frac{1}{2}\left(y-5\right)=\frac{1}{4}\left(y-1\right).\)

### Solution

Distribute. | |

Simplify. | |

Multiply by the LCD, 4. | |

Distribute. | |

Simplify. | |

Collect the \(y\) terms to the left. | |

Simplify. | |

Collect the constants to the right. | |

Simplify. | |

Check: Substitute \(9\) for \(y.\) | |