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.

This figure has two columns. The first column has the equation x plus 4 equals 12. Underneath there is x plus 4 minus 4 equals 12 minus 4. Under this there is x equals 8. The second column has the equation y minus 5 equals 9. Underneath there is the equation y minus 5 plus 5 equals 9 plus 5. Under this there is y equals 14.

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

Properties of Inequalities

Subtraction Property of Equality	Addition 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.

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