Mathematics » Introducing Integers » Solve Equations Using Integers, The Division Property of Equality

# Solving Equations with Integers Using the Addition and Subtraction Properties of Equality

## Solving Equations with Integers Using the Addition and Subtraction Properties of Equality

In Mathematics 102, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

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

### Properties of Inequalities

Subtraction Property of EqualityAddition Property of Equality
$$\text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,$$

$$\text{if}\phantom{\rule{0.2em}{0ex}}a=b\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}a-c=b-c.$$

$$\text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,$$

$$\text{if}\phantom{\rule{0.2em}{0ex}}a=b\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}a+c=b+c.$$

## Example

Solve: $$y+9=5.$$

### Solution

 Subtract 9 from each side to undo the addition. Simplify.

Check the result by substituting $$-4$$ into the original equation.

 $$y+9=5\phantom{\rule{1.4em}{0ex}}$$ Substitute −4 for y $$-4+9\stackrel{?}{=}5\phantom{\rule{1.4em}{0ex}}$$ $$5=5✓$$

Since $$y=-4$$ makes $$y+9=5$$ a true statement, we found the solution to this equation.

## Example

Solve: $$a-6=-8$$

### Solution

 Add 6 to each side to undo the subtraction. Simplify. Check the result by substituting $$-2$$ into the original equation: Substitute $$-2$$ for $$a$$

The solution to $$a-6=-8$$ is $$-2.$$

Since $$a=-2$$ makes $$a-6=-8$$ a true statement, we found the solution to this equation.