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\(=\)
\(a\)
\(+\)
\(b\)

Properties for
Equivalent Expressions

Learn to use the distributive, associative, and commutative properties to rewrite and simplify algebraic expressions.

Pre-Algebra

What Are Equivalent Expressions?

Two expressions are equivalent if they produce the same value for every possible value of the variable. They may look different, but they always simplify to the same result.

\(2(x+3)\)
\(2x+6\)
\(=\)

Why Do Properties Matter?

Properties are the rules that let you rearrange and simplify expressions while keeping them equivalent. Think of them as tools in your algebra toolkit.

Simplify
Combine like terms and reduce complex expressions to simpler forms.
Solve
Rewrite equations in forms that make solving for the variable easier.
Verify
Prove that two different-looking expressions are actually equal.

The Three Key Properties

Commutative
Change the order of terms.
\( a + b = b + a \)
\( a \cdot b = b \cdot a \)
Associative
Change the grouping of terms.
\( (a+b)+c = a+(b+c) \)
\( (ab)c = a(bc) \)
Distributive
Multiply across a sum or difference.
\( a(b+c) = ab + ac \)

The Commutative Property

ADDITION
\(3 + 7\)
\(=\)
\(7 + 3\)
 MULTIPLICATION
\(5 \times 4\)
\(=\)
\(4 \times 5\)

The Associative Property

ADDITION
\((2+5)+3\)
\(=\)
\(2+(5+3)\)
 MULTIPLICATION
\((4 \times 2) \times 6\)
\(=\)
\(4 \times (2 \times 6)\)

The Distributive Property

\(a(b+c)\)
 DISTRIBUTE
\(a \cdot b\)
\(+\)
\(a \cdot c\)

Comparing the Properties

What Changes? Operations Example
Commutative Order \( +, \times \) \( 3 + 5 = 5 + 3 \)
Associative Grouping \( +, \times \) \( (2+4)+1 = 2+(4+1) \)
Distributive Form \( \times \) over \( + \) \( 3(x+2) = 3x+6 \)

Where They Don't Apply

The commutative and associative properties do not work for subtraction or division. Changing order or grouping with these operations changes the result.

Subtraction
\( 7 - 3 = 4 \)
but \( 3 - 7 = -4 \)
Order matters!
Division
\( 12 \div 4 = 3 \)
but \( 4 \div 12 = \frac{1}{3} \)
Order matters!

How to Rewrite Expressions

1
Identify the operation. Is the expression using addition, multiplication, or both?
2
Choose the right property. Reorder? Use commutative. Regroup? Use associative. Expand or factor? Use distributive.
3
Apply the property. Rewrite the expression following the property's rule.
4
Simplify if possible. Combine like terms or compute arithmetic to get the final form.
5
Verify. Plug in a test value to confirm both expressions give the same result.

Example 1: Distributive Property

Expand and simplify: \( 4(x + 5) \)

\(4(x+5)\)
\(4 \cdot x + 4 \cdot 5\)
\(4x + 20\)
 Original Distribute Simplified

Example 2: Commutative + Associative

Simplify: \( 3 + 7x + 12 + 5x \)

1
Commutative property: Rearrange terms so like terms are next to each other.
\( 7x + 5x + 3 + 12 \)
2
Associative property: Group the like terms.
\( (7x + 5x) + (3 + 12) \)
3
Combine like terms: Add the coefficients and constants.
\( 12x + 15 \)

Example 3: All Three Properties

Simplify: \( 2(3x + 4) + 5x \)

1
Distributive: Expand \( 2(3x + 4) \).
\( 6x + 8 + 5x \)
2
Commutative: Rearrange to group like terms.
\( 6x + 5x + 8 \)
3
Combine like terms: Add the \( x \)-terms.
\( 11x + 8 \)

Distributive in Reverse: Factoring

The distributive property also works backwards. If you see a common factor in each term, you can factor it out.

\(6x + 18\)
 Expanded
Factor out \(6\)
\(6(x+3)\)
 Factored

Common Mistakes

✕ Incomplete Distribution
Writing \( 3(x + 4) = 3x + 4 \) instead of \( 3x + 12 \). You must multiply the factor by every term inside.
✕ Subtracting Non-Commutatively
Thinking \( 5 - 3 = 3 - 5 \). The commutative property does not apply to subtraction.
✕ Combining Unlike Terms
Adding \( 3x + 5 = 8x \). The constant \( 5 \) and the variable term \( 3x \) are not like terms.

Summary of Properties

Property Rule In Words Example
Commutative (+) \( a + b = b + a \) Swap order \( 4 + 9 = 9 + 4 \)
Commutative (×) \( ab = ba \) Swap order \( 3 \cdot 7 = 7 \cdot 3 \)
Associative (+) \( (a+b)+c = a+(b+c) \) Move grouping \( (1+2)+3 = 1+(2+3) \)
Associative (×) \( (ab)c = a(bc) \) Move grouping \( (2 \cdot 5) \cdot 3 = 2 \cdot (5 \cdot 3) \)
Distributive \( a(b+c) = ab+ac \) Multiply across \( 5(x+2) = 5x+10 \)

Question 1 of 5

Which expression is equivalent to \( 5(2x + 3) \)?

Question 2 of 5

Which property justifies that \( 8 + 3x = 3x + 8 \)?

Question 3 of 5

Simplify: \( 4x + 7 + 2x + 3 \)

Question 4 of 5

Which expression is the factored form of \( 12x + 8 \)?

Question 5 of 5

Which property is used to rewrite \( (6 \cdot x) \cdot 5 \) as \( 6 \cdot (x \cdot 5) \)?