## Solving Equations with Decimal Coefficients

Contents

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, \(0.3=\frac{3}{10}\) and \(0.17=\frac{17}{100}.\) So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the **least common denominator**.

## Example

Solve: \(0.8x-5=7.\)

### Solution

The only decimal in the equation is \(0.8.\) Since \(0.8=\frac{8}{10},\) the LCD is \(10.\) We can multiply both sides by \(10\) to clear the decimal.

Multiply both sides by the LCD. | |

Distribute. | |

Multiply, and notice, no more decimals! | |

Add 50 to get all constants to the right. | |

Simplify. | |

Divide both sides by 8. | |

Simplify. | |

Check: Let \(x=15.\) | |

## Example

Solve: \(0.06x+0.02=0.25x-1.5.\)

### Solution

Look at the decimals and think of the equivalent fractions.

\(0.06=\frac{6}{100},\phantom{\rule{1em}{0ex}}0.02=\frac{2}{100},\phantom{\rule{1em}{0ex}}0.25=\frac{25}{100},\phantom{\rule{1em}{0ex}}1.5=1\frac{5}{10}\)

Notice, the LCD is \(100.\)

By multiplying by the LCD we will clear the decimals.

Multiply both sides by 100. | |

Distribute. | |

Multiply, and now no more decimals. | |

Collect the variables to the right. | |

Simplify. | |

Collect the constants to the left. | |

Simplify. | |

Divide by 19. | |

Simplify. | |

Check: Let \(x=8.\) | |

The next example uses an equation that is typical of the ones we will see in the money applications in the next tutorial. Notice that we will distribute the decimal first before we clear all decimals in the equation.

## Example

Solve: \(0.25x+0.05\left(x+3\right)=2.85.\)

### Solution

Distribute first. | |

Combine like terms. | |

To clear decimals, multiply by 100. | |

Distribute. | |

Subtract 15 from both sides. | |

Simplify. | |

Divide by 30. | |

Simplify. | |

Check: Let \(x=9.\) | |

### Optional Video: Solving Multiple Step Equations Involving Decimals

## Key Concepts

**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.

- Find the least common denominator of