How to Use Properties of Numbers to Write Equivalent Expressions?

The key properties of arithmetic — commutative, associative, and distributive — let you rewrite expressions in different but equivalent forms. Knowing how to apply these properties is essential for simplifying expressions and solving equations on the GED Math test. This lesson explains each property and shows you how to use them to generate equivalent expressions.

What Properties Are Used to Write Equivalent Expressions?

Three properties allow you to rewrite an expression without changing its value:

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• Commutative Property — order of addition or multiplication does not matter.
• Associative Property — grouping of addition or multiplication does not matter.
• Distributive Property — multiplying a sum or difference by a factor distributes across each term.

The Three Key Properties

1. Commutative Property

• Addition: \(\color{blue}{a + b = b + a}\)  →  \(\color{blue}{3x + 5 = 5 + 3x}\)
• Multiplication: \(\color{blue}{a \times b = b \times a}\)  →  \(\color{blue}{4 \times y = y \times 4}\)

2. Associative Property

• Addition: \(\color{blue}{(a + b) + c = a + (b + c)}\)  →  \(\color{blue}{(2x + 3) + 7 = 2x + (3 + 7) = 2x + 10}\)
• Multiplication: \(\color{blue}{(a \times b) \times c = a \times (b \times c)}\)  →  \(\color{blue}{(3 \times x) \times 4 = 3 \times (x \times 4) = 12x}\)

3. Distributive Property

\(\color{blue}{a(b + c) = \text{ ab } + \text{ ac }}\)

• \(\color{blue}{3(x + 4) = 3x + 12}\)
• \(\color{blue}{5(2y – 3) = 10y – 15}\)
• \(\color{blue}{-2(4n + 1) = -8n – 2}\)

The distributive property can also be used in reverse (factoring): \(\color{blue}{6x + 9 = 3(2x + 3)}\).

Step-by-Step Summary

1. Identify which property applies: commutative (reordering), associative (regrouping), or distributive (expanding/factoring).
2. Apply the property to rewrite the expression.
3. Combine any like terms to reach the simplest equivalent form.
4. Verify by substituting a test value into both forms to confirm they are equal.

Watch: Simplifying Expressions Using Properties (Khan Academy)

Sal Khan shows how to use the distributive property and combining like terms together:

Worked Examples

Example 1: Use the commutative property to rewrite \(\color{blue}{7 + 3x}\).

\(\color{blue}{7 + 3x = 3x + 7}\)  (reorder the terms; value is unchanged)

Example 2: Use the distributive property to expand \(\color{blue}{4(2x – 5)}\).

\(\color{blue}{4 \times 2x – 4 \times 5 = 8x – 20}\)

Example 3: Use properties to simplify \(\color{blue}{3(x + 2) + 5x}\).

Distribute: \(\color{blue}{3x + 6 + 5x}\). Combine like terms: \(\color{blue}{8x + 6}\).

Example 4: Rewrite \(\color{blue}{2x + 6}\) using the distributive property in reverse (factor).

GCF of 2x and 6 is 2: \(\color{blue}{2x + 6 = 2(x + 3)}\).

More Practice: The Distributive Property Step-by-Step (Math with Mr. J)

Math with Mr. J walks through expanding and factoring with the distributive property:

Exercises

1. Use the commutative property to rewrite \(\color{blue}{9 + 4n}\).
2. Use the distributive property to expand \(\color{blue}{6(x + 3)}\).
3. Simplify \(\color{blue}{2(3y + 4) + y}\) using properties.
4. Factor \(\color{blue}{10x + 15}\) using the distributive property.
5. Use the associative property to regroup and simplify \(\color{blue}{(5 + 3x) + 2x}\).
6. Expand and simplify \(\color{blue}{4(a – 2) + 3a}\).

Answers

1. \(\color{blue}{4n + 9}\)
2. \(\color{blue}{6x + 18}\)
3. \(\color{blue}{6y + 8 + y = 7y + 8}\)
4. \(\color{blue}{5(2x + 3)}\)
5. \(\color{blue}{5 + (3x + 2x) = 5 + 5x}\)
6. \(\color{blue}{4a – 8 + 3a = 7a – 8}\)

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Frequently Asked Questions

What is the most useful property for rewriting expressions?

The distributive property is the most powerful. It lets you expand a factored expression into a sum of terms, or reverse the process to factor out a common factor — both very common GED Math tasks.

Does the commutative property work for subtraction or division?

No. The commutative property applies only to addition and multiplication. Subtraction and division are not commutative: \(\color{blue}{5 – 3 \ne 3 – 5}\), and \(\color{blue}{8 \div 2 \ne 2 \div 8}\).

How do I know which property to use?

Look at what you need to do: reordering terms → commutative; regrouping without reordering → associative; removing or adding parentheses involving multiplication → distributive.

Related Topics

• Identify Equivalent Expressions
• Use Algebra Tiles to Identify Equivalent Expressions
• Using Distributive Property to Factor Variable Expressions
• Identifying Expressions and Equations
• Using Area Models to Factor Variable Expressions