Equivalent Expressions - Seventh Grade

Properties, Combining Like Terms, Distributive Property & Factoring

1. Properties of Addition and Multiplication

Commutative Property

Order doesn't matter!

Addition: a + b = b + a

Example: 5 + 3 = 3 + 5 = 8

Example: x + 7 = 7 + x

Multiplication: a × b = b × a

Example: 4 × 6 = 6 × 4 = 24

Example: 3x = x × 3

⚠️ Important: Subtraction and division are NOT commutative!

• 5 − 3 ≠ 3 − 5

• 10 ÷ 2 ≠ 2 ÷ 10

Associative Property

Grouping doesn't matter!

Addition: (a + b) + c = a + (b + c)

Example: (2 + 3) + 4 = 2 + (3 + 4) = 9

Multiplication: (a × b) × c = a × (b × c)

Example: (2 × 3) × 4 = 2 × (3 × 4) = 24

Identity Property

Additive Identity: a + 0 = a

Adding zero doesn't change the value

Example: x + 0 = x

Multiplicative Identity: a × 1 = a

Multiplying by one doesn't change the value

Example: 5x × 1 = 5x

Zero Property of Multiplication

a × 0 = 0

Any number multiplied by zero equals zero

2. Distributive Property

Formula

a(b + c) = ab + ac

a(b − c) = ab − ac

How it Works

Multiply the number outside the parentheses

by EACH term inside the parentheses

Example 1: With Numbers

Simplify: 3(4 + 5)

Method 1: Add first, then multiply

3(4 + 5) = 3(9) = 27

Method 2: Use distributive property

3(4 + 5) = 3(4) + 3(5)

= 12 + 15 = 27

Answer: 27

Example 2: With Variables

Simplify: 4(x + 3)

Distribute 4 to both x and 3:

4(x) + 4(3)

= 4x + 12

Answer: 4x + 12

Example 3: With Subtraction

Simplify: 5(2x − 4)

5(2x) − 5(4)

= 10x − 20

Answer: 10x − 20

Example 4: Negative Outside

Simplify: −3(x + 5)

−3(x) + (−3)(5)

= −3x − 15

Answer: −3x − 15

3. Like Terms

Definition

Like terms have the SAME variable(s)

with the SAME exponent(s)

Coefficients can be different!

Examples of Like Terms

Like Terms ✓	NOT Like Terms ✗
3x and 5x	3x and 3y
4y² and −2y²	4x² and 4x
7ab and ab	5ab and 5a
6 and −9 (constants)	x³ and x²

4. Combining Like Terms

Rule

Add or subtract the COEFFICIENTS

Keep the variable part the same

Steps

Step 1: Identify like terms

Step 2: Group like terms together

Step 3: Add or subtract coefficients

Step 4: Write the simplified expression

Example 1: Simple

Simplify: 3x + 5x

Both terms have the same variable (x)

Add coefficients: 3 + 5 = 8

Keep variable: x

Answer: 8x

Example 2: Multiple Terms

Simplify: 4x + 2y + 7x − 5y

Step 1: Group like terms

(4x + 7x) + (2y − 5y)

Step 2: Combine each group

11x + (−3y)

= 11x − 3y

Answer: 11x − 3y

Example 3: With Exponents

Simplify: 5x² + 3x + 2x² − 7x

Group like terms:

(5x² + 2x²) + (3x − 7x)

= 7x² + (−4x)

= 7x² − 4x

Answer: 7x² − 4x

5. Adding and Subtracting Linear Expressions

Adding Expressions

Problem: (3x + 5) + (2x + 7)

Step 1: Remove parentheses

3x + 5 + 2x + 7

Step 2: Combine like terms

(3x + 2x) + (5 + 7)

= 5x + 12

Answer: 5x + 12

Subtracting Expressions

⚠️ Important: When subtracting, distribute the negative sign!

Change the sign of EVERY term in the second expression

Problem: (5x + 8) − (2x + 3)

Step 1: Distribute the negative

5x + 8 − 2x − 3

Step 2: Combine like terms

(5x − 2x) + (8 − 3)

= 3x + 5

Answer: 3x + 5

6. Factoring Linear Expressions

What is Factoring?

Factoring is the REVERSE of the distributive property

Write a sum as a product

Find the Greatest Common Factor (GCF) and "factor it out"

Steps to Factor

Step 1: Find the GCF of all terms

Step 2: Divide each term by the GCF

Step 3: Write as: GCF(remaining terms)

Example 1: Numbers Only

Factor: 12 + 18

Step 1: GCF of 12 and 18 is 6

Step 2: Divide: 12÷6=2, 18÷6=3

Step 3: Write as product

Answer: 6(2 + 3)

Example 2: With Variables

Factor: 6x + 9

Step 1: GCF of 6 and 9 is 3

Step 2: Divide: 6x÷3=2x, 9÷3=3

Step 3: 3(2x + 3)

Answer: 3(2x + 3)

Example 3: Variable as GCF

Factor: 5x + 10x²

GCF = 5x (both number and variable)

5x ÷ 5x = 1

10x² ÷ 5x = 2x

Answer: 5x(1 + 2x)

7. Identifying Equivalent Expressions

What are Equivalent Expressions?

Expressions that have the SAME VALUE

for all values of the variable(s)

Example: 2x + 4 and 2(x + 2) are equivalent

How to Check if Expressions are Equivalent

Method 1: Simplify both expressions

If they simplify to the same form, they're equivalent

Method 2: Substitute the same value

If they give the same result, they might be equivalent

(Try multiple values to be sure!)

Example

Are 3(x + 2) and 3x + 6 equivalent?

Method 1: Simplify first expression

3(x + 2) = 3x + 6

Both expressions are the same!

Method 2: Test with x = 1

3(1 + 2) = 3(3) = 9

3(1) + 6 = 3 + 6 = 9

Yes, they are equivalent!

Quick Reference: Key Formulas

Property/Concept	Formula
Commutative (Addition)	a + b = b + a
Commutative (Multiplication)	a × b = b × a
Associative (Addition)	(a + b) + c = a + (b + c)
Associative (Multiplication)	(a × b) × c = a × (b × c)
Distributive Property	a(b + c) = ab + ac
Identity (Addition)	a + 0 = a
Identity (Multiplication)	a × 1 = a
Zero Property	a × 0 = 0