## Solving Equations with Variables and Constants on Both Sides

Contents

The next example will be the first to have variables *and* constants on both sides of the equation. As we did before, we’ll collect the **variable** terms to one side and the **constants** to the other side.

## Example

Solve: \(7x+5=6x+2.\)

### Solution

Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are \(7x\) and \(6x.\) Since \(7\) is greater than \(6,\) make the left side the variable side and so the right side will be the constant side.

Collect the variable terms to the left side by subtracting \(6x\) from both sides. | |

Simplify. | |

Now, collect the constants to the right side by subtracting 5 from both sides. | |

Simplify. | |

The solution is \(x=-3\). | |

Check: Let \(x=-3\). | |

We’ll summarize the steps we took so you can easily refer to them.

### How to Solve an equation with variables and constants on both sides.

- Choose one side to be the variable side and then the other will be the constant side.
- Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
- Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable \(1,\) using the Multiplication or Division Property of Equality.
- Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

## Example

Solve: \(6n-2=-3n+7.\)

### Solution

We have \(6n\) on the left and \(-3n\) on the right. Since \(6>-3,\) make the left side the “variable” side.

We don’t want variables on the right side—add \(3n\) to both sides to leave only constants on the right. | |

Combine like terms. | |

We don’t want any constants on the left side, so add 2 to both sides. | |

Simplify. | |

The variable term is on the left and the constant term is on the right. To get the coefficient of \(n\) to be one, divide both sides by 9. | |

Simplify. | |

Check: Substitute 1 for \(n\). | |

## Example

Solve: \(2a-7=5a+8.\)

### Solution

This equation has \(2a\) on the left and \(5a\) on the right. Since \(5>2,\) make the right side the variable side and the left side the constant side.

Subtract \(2a\) from both sides to remove the variable term from the left. | |

Combine like terms. | |

Subtract 8 from both sides to remove the constant from the right. | |

Simplify. | |

Divide both sides by 3 to make 1 the coefficient of \(a\). | |

Simplify. | |

Check: Let \(a=-5\). | |

Note that we could have made the left side the variable side instead of the right side, but it would have led to a negative coefficient on the variable term. While we could work with the negative, there is less chance of error when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we still follow the same steps to get the **solution**.

## Example

Solve: \(\frac{3}{2}\phantom{\rule{0.1em}{0ex}}x+5=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x-3.\)

### Solution

Since \(\frac{3}{2}>\frac{1}{2},\) make the left side the variable side and the right side the constant side.

Subtract \(\frac{1}{2}x\) from both sides. | |

Combine like terms. | |

Subtract 5 from both sides. | |

Simplify. | |

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

We follow the same steps when the equation has decimals, too.

## Example

Solve: \(3.4x+4=1.6x-5.\)

### Solution

Since \(3.4>1.6,\) make the left side the variable side and the right side the constant side.

Subtract \(1.6x\) from both sides. | |

Combine like terms. | |

Subtract 4 from both sides. | |

Simplify. | |

Use the Division Property of Equality. | |

Simplify. | |

Check: Let \(x=-5\). | |