Mathematics » Introducing Decimals » Solve Equations with Decimals

# Solving Equations with Decimals

## 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 Equality For 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 Equality For 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 Equality For 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 Equality For 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.

## 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.

## 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.

## 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 $$p=11.7.$$