How to Use Properties of Numbers to Write Equivalent Expressions?

How to use properties of numbers to write equivalent expressions, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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:

• 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

Why can you use properties of operations to write equivalent expressions?

Each property describes a change that never alters an expression’s value, for any value of the variable. Swapping the order (commutative), regrouping (associative) or distributing a factor all keep the result the same. For example, (2x + 3) + 7 and 2x + 10 both equal 16 when x = 3 and 30 when x = 10. Because the value never changes, the new form is equivalent.

How do you use the distributive property to write equivalent expressions?

Multiply the factor outside the parentheses by every term inside. 4(x + 5) becomes 4x + 20, and 5(2y − 3) becomes 10y − 15. It also works in reverse as factoring: 12x + 18 = 6(2x + 3), because 6 is the greatest common factor. A frequent slip on GED Math is multiplying only the first term, so check each term inside the parentheses.

How do you write two equivalent expressions for 2(2y − 4)?

Apply the distributive property first to get 4y − 8. For a second form, use the commutative property to write −8 + 4y, or factor out 4 to get 4(y − 2). All three agree for every value of y. With y = 5: 2(2 × 5 − 4) = 12, 4(5) − 8 = 12 and 4(5 − 2) = 12.

How do you write an equivalent expression by combining like terms?

Use the commutative and associative properties to group like terms together, then add their coefficients. In 3x + 7 + 5x − 2, reorder to (3x + 5x) + (7 − 2), which gives 8x + 5. Only terms with the same variable and exponent combine, so 8x and 5 stay separate, and x² terms never merge with x terms.

How do you use properties of exponents to write an equivalent expression?

Apply the exponent rules, which keep the value unchanged: multiplying powers of the same base adds the exponents, dividing subtracts them, and a power of a power multiplies them. For example, y² × y⁵ = y⁷ and (2x³)² = 2² × x⁶ = 4x⁶. Combined with the commutative property, 3a² × 4a = 12a³ once you multiply the coefficients 3 × 4.

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