Mathematics » Properties of Real Numbers » Distributive Property

Evaluating Expressions Using the Distributive Property

Evaluating Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part , we will evaluate the form with parentheses, and in part we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example

When \(y=10\) evaluate: \(\phantom{\rule{0.2em}{0ex}}6\left(5y+1\right)\) \(\phantom{\rule{0.2em}{0ex}}6·5y+6·1.\)

Solution

  
 \(6\left(5y+1\right)\)
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Simplify in the parentheses.\(6\left(51\right)\)
Multiply.\(306\)
  
 Evaluating Expressions Using the Distributive Property
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Simplify.Evaluating Expressions Using the Distributive Property
Add.Evaluating Expressions Using the Distributive Property

Notice, the answers are the same. When \(y=10,\)

\(6\left(5y+1\right)=6·5y+6·1.\)

Try it yourself for a different value of \(y.\)

Example

When \(y=3,\) evaluate \(\phantom{\rule{0.2em}{0ex}}-2\left(4y+1\right)\)\(\phantom{\rule{0.2em}{0ex}}-2·4y+\left(-2\right)·1.\)

Solution

  
 \(\phantom{\rule{0.2em}{0ex}}-2\left(4y+1\right)\)
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Simplify in the parentheses.\(-2\left(13\right)\)
Multiply.\(-26\)
  
 \(\phantom{\rule{0.2em}{0ex}}-2·4y+\left(-2\right)·1\)
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Multiply.\(-24-2\)
Subtract.\(-26\)
The answers are the same. When \(y=3,\)\(-2\left(4y+1\right)=-8y-2\)

Example

When \(y=35\) evaluate \(\phantom{\rule{0.2em}{0ex}}\text{−}\left(y+5\right)\) and \(\phantom{\rule{0.2em}{0ex}}\text{−}\mathit{\text{y}}-5\) to show that \(-\left(y+5\right)=\text{−}\mathit{\text{y}}-5.\)

Solution

  
 \(\phantom{\rule{0.2em}{0ex}}\text{−}\left(y+5\right)\)
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Add in the parentheses.\(-\left(40\right)\)
Simplify.\(-40\)
  
 \(\phantom{\rule{0.2em}{0ex}}-y-5\)
Evaluating Expressions Using the Distributive PropertyEvaluating Expressions Using the Distributive Property
Simplify.\(-40\)
The answers are the same when \(y=35,\) demonstrating that\(-\left(y+5\right)=\text{−}y-5\)

Key Concepts

  • Distributive Property:
    • If \(a,b,c\) are real numbers then
      • \(a\left(b+c\right)=ab+ac\)
      • \(\left(b+c\right)a=ba+ca\)
      • \(a\left(b\cdot c\right)=ab\cdot ac\)

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