Topics
5. Algebraic Expressions
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United States of America
Grade 7

5.05 The distributive property with rational numbers

Lesson
Practice

Introduction

Let us recall that the expression a(b+c)=ab+ac by the distributive property of equality.

Since the distributive property is true for all numbers, we can also apply it to expressions with rational numbers.

The distributive property with rational numbers

Whether algebraic expressions contain integers or rational numbers, we need to use the distributive property if we see any parentheses.

We now know how to combine like terms, perform operations with positive or negative algebraic terms, get the least common multiple of the denominator for fractions, and find the lowest terms. This means we can now use all these skills to simplify expressions involving the four operations and the distributive property with rational numbers.

When there is an expression involving a mixture of addition, subtraction, multiplication, and division, as well as distributing terms in parentheses, we need to follow the correct order of operations:

  1. If we see any parentheses, we need to distribute them first.

  2. Collect the like terms to simplify.

  3. For expressions involving fractions with different denominators, rename the fractions so that they have the same denominators. Get the least common multiple of the denominator and find the equivalent fractions before adding them.

  4. Make sure all the fractions are in their lowest terms when asked to simplify the expressions.

Examples

Example 1

Expand - 6.1 \left(3.5y + 4.5\right).

Worked Solution
Create a strategy

Use the distributive property and observe the operations with negative signs.

Apply the idea

To expand the expression:

\displaystyle - 6.1 \left(3.5y + 4.5\right)\displaystyle =\displaystyle (-6.1) \times 3.5y + (-6.1)\times 4.5Use the distributive property
\displaystyle =\displaystyle -21.35y + (-27.45)Evaluate the multiplication
\displaystyle =\displaystyle -21.35y - 27.45Multiply the positive sign to negative number

Example 2

Distribute and simplify: \dfrac{1}{4}(x-4)-\dfrac{1}{5}x

Worked Solution
Create a strategy

Expand the expressions, combine like terms, and make sure the fractions are in lowest terms when simplified.

Apply the idea
\displaystyle \dfrac{1}{4}(x-4)-\dfrac{1}{5}x\displaystyle =\displaystyle \dfrac{1}{4}x-\dfrac{4}{4}-\dfrac{1}{5}xUse the distributive property
\displaystyle =\displaystyle \dfrac{1}{4}x-\dfrac{1}{5}x-1Use commutative property and simplify fraction
\displaystyle =\displaystyle \dfrac{5}{20}x-\dfrac{4}{20}x-1Rename fractions to make their denominators similar
\displaystyle =\displaystyle \dfrac{1}{20}x-1Combine like terms
Reflect and check

\dfrac{1}{20}x-1 can also be written as \dfrac{x}{20}-1. Always check if fractions are in their lowest terms.

Idea summary

The distributive property can be used as strategy to add, subtract, factor, and expand linear expressions with rational coefficients.

To simplify an expression containing a mixture of operations:

  • Expand the expression with parentheses.

  • Combine the like terms to simplify.

  • For expressions involving fractions, add fractions with the same denominator. Find the least common denominator and rename the fractions if needed before adding or subtracting them.

  • Make sure all fractions are in their lowest terms.

Outcomes

7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

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