## Solving Equations with Decimals

In previous tutorials, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

### Properties of Equality

Subtraction Property of EqualityFor any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\) If \(a=b,\) then \(a-c=b-c.\) | Addition Property of EqualityFor any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\) If \(a=b,\) then \(a+c=b+c.\) |

The Division Property of EqualityFor any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\ne 0\) If \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}\) | The Multiplication Property of EqualityFor any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\) If \(a=b,\) then \(ac=bc\) |

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

## Example

Solve: \(y+2.3=-4.7.\)

### Solution

We will use the **Subtraction Property of Equality** to isolate the variable.

Simplify. | ||

Check: | ||

Simplify. |

Since \(y=-7\) makes \(y+2.3=-4.7\) a true statement, we know we have found a solution to this equation.

### Optional Video: Solve a One Step Equation With Decimals by Adding and Subtracting

## Example

Solve: \(a-4.75=-1.39.\)

### Solution

We will use the Addition Property of Equality.

Add 4.75 to each side, to undo the subtraction. | ||

Simplify. | ||

Check: | ||

Since the result is a true statement, \(a=3.36\) is a solution to the equation.

### Optional Video: Solve a One Step Equation With Decimals by Multiplying

## Example

Solve: \(-4.8=0.8n.\)

### Solution

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for \(n.\)

We must divide both sides by 0.8 to isolate n. | ||

Simplify. | ||

Check: | ||

Since \(n=-6\) makes \(-4.8=0.8n\) a true statement, we know we have a solution.

### Optional Video: Solve a One Step Equation With Decimals by Dividing

## Example

Solve: \(\frac{p}{-1.8}=-6.5.\)

### Solution

We will use the **Multiplication Property of Equality**.

Here, p is divided by −1.8. We must multiply by −1.8 to isolate p | ||

Multiply. | ||

Check: | ||

A solution to \(\frac{p}{-1.8}=-6.5\) is \(p=11.7.\)