Mathematics » Mathematics 103: Integers » Solve Equations Using Integers; The Division Property of Equality

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

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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.

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

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

 Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Subtract 9 from each side to undo the addition.Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Simplify.Solving Equations with Integers Using the Addition and Subtraction Properties of Equality

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

 Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Add 6 to each side to undo the subtraction.Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Simplify.Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Check the result by substituting \(-2\) into the original equation:Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
Substitute \(-2\) for \(a\)Solving Equations with Integers Using the Addition and Subtraction Properties of Equality
 Solving Equations with Integers Using the Addition and Subtraction Properties of Equality

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.

Optional Video: Solving One Step Equation by Adding or Subtracting Integers

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