1. Properties of Addition and Multiplication

Commutative Property:

Definition: The order of numbers does not change the result.

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

Example: \( 3 + 5 = 5 + 3 = 8 \)

With variables: \( x + 7 = 7 + x \)

Multiplication: \( a \times b = b \times a \) or \( ab = ba \)

Example: \( 4 \times 6 = 6 \times 4 = 24 \)

With variables: \( 5y = y \cdot 5 \)

Note: Subtraction and division are NOT commutative!

Associative Property:

Definition: The grouping of numbers does not change the result.

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

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

With variables: \( (x + 2) + 5 = x + (2 + 5) = x + 7 \)

Multiplication: \( (a \times b) \times c = a \times (b \times c) \)

Example: \( (2 \times 3) \times 4 = 2 \times (3 \times 4) = 24 \)

With variables: \( (3x) \times 4 = 3 \times (4x) = 12x \)

Distributive Property:

Definition: Multiply a sum by multiplying each addend separately, then add.

\( a(b + c) = ab + ac \)

\( a(b - c) = ab - ac \)

Example: \( 3(x + 4) = 3x + 12 \)

Example: \( 5(2y - 3) = 10y - 15 \)

Identity Properties:

• Additive Identity: \( a + 0 = a \) (Adding 0 doesn't change the value)
• Multiplicative Identity: \( a \times 1 = a \) (Multiplying by 1 doesn't change the value)

Zero Property of Multiplication:

\( a \times 0 = 0 \) (Any number multiplied by 0 equals 0)

2. Multiply Using the Distributive Property

Basic Formula:

\( a(b + c) = ab + ac \)

Steps to Distribute:

1. Multiply the term outside the parentheses by each term inside
2. Keep the operation signs (+ or −) the same
3. Simplify if possible

Examples:

Example 1: \( 4(x + 3) \)

\( = 4 \cdot x + 4 \cdot 3 = 4x + 12 \)

Example 2: \( -2(y - 5) \)

\( = -2 \cdot y - (-2) \cdot 5 = -2y + 10 \)

Example 3: \( 6(2a + 3b - 4) \)

\( = 6 \cdot 2a + 6 \cdot 3b + 6 \cdot (-4) \)

\( = 12a + 18b - 24 \)

Example 4: \( \frac{1}{2}(8x - 4) \)

\( = \frac{1}{2} \cdot 8x - \frac{1}{2} \cdot 4 = 4x - 2 \)

Negative Signs:

When distributing a negative sign, change all signs inside:

\( -(3x - 7) = -1(3x - 7) = -3x + 7 \)

Reverse Distribution (Factoring):

Going backwards: \( ab + ac = a(b + c) \)

Example: \( 3x + 9 = 3(x + 3) \)

3. Write Equivalent Expressions Using Properties

Definition: Equivalent expressions have the same value for all values of the variables, but are written differently.

Using Commutative Property:

\( 5 + x \equiv x + 5 \)

\( 3y \equiv y \cdot 3 \)

Using Associative Property:

\( (x + 3) + 5 \equiv x + (3 + 5) \equiv x + 8 \)

\( 2(3x) \equiv (2 \cdot 3)x \equiv 6x \)

Using Distributive Property:

\( 4(x + 2) \equiv 4x + 8 \)

\( 3(2y - 5) \equiv 6y - 15 \)

Combining Multiple Properties:

Example 1: Simplify \( 2(x + 3) + 4x \)

Step 1: Distribute: \( 2x + 6 + 4x \)

Step 2: Combine like terms: \( 6x + 6 \)

So \( 2(x + 3) + 4x \equiv 6x + 6 \)

Example 2: Show that \( 5(2n + 1) - 3n \) is equivalent to \( 7n + 5 \)

Distribute: \( 10n + 5 - 3n \)

Combine like terms: \( 7n + 5 \) ✓

Testing for Equivalence:

To verify expressions are equivalent, substitute the same value and check if results match:

Test \( 3(x + 2) \) and \( 3x + 6 \) when \( x = 4 \)

\( 3(4 + 2) = 3(6) = 18 \)

\( 3(4) + 6 = 12 + 6 = 18 \) ✓ Equivalent!

4. Add and Subtract Like Terms

Definition: Like terms have the same variable(s) raised to the same power(s).

Identifying Like Terms:

Rules for Combining Like Terms:

• Add or subtract the coefficients (numbers in front)
• Keep the variable part the same
• You cannot combine unlike terms

Formula:

\( ax + bx = (a + b)x \)

\( ax - bx = (a - b)x \)

Examples:

Example 1: \( 5x + 3x \)

\( = (5 + 3)x = 8x \)

Example 2: \( 7y - 4y \)

\( = (7 - 4)y = 3y \)

Example 3: \( 2a^2 + 5a^2 - 3a^2 \)

\( = (2 + 5 - 3)a^2 = 4a^2 \)

Example 4: Simplify \( 4x + 3y + 2x - y \)

Group like terms: \( (4x + 2x) + (3y - y) \)

\( = 6x + 2y \)

Example 5: Simplify \( 3m + 7 - m + 4 \)

Group like terms: \( (3m - m) + (7 + 4) \)

\( = 2m + 11 \)

Important: When a variable has no visible coefficient, it's understood to be 1 or -1.

\( x = 1x \) and \( -x = -1x \)

5. Add and Subtract Linear Expressions

Definition: A linear expression is an algebraic expression where the variable has an exponent of 1.

Examples: \( 3x + 5 \), \( 7y - 2 \), \( 4a + 3b - 1 \)

Steps to Add Linear Expressions:

1. Remove parentheses if present (distribute any coefficients)
2. Identify and group like terms
3. Combine coefficients of like terms
4. Write in simplified form

Addition Examples:

Example 1: Add \( (3x + 5) + (2x + 7) \)

Remove parentheses: \( 3x + 5 + 2x + 7 \)

Group like terms: \( (3x + 2x) + (5 + 7) \)

Combine: \( 5x + 12 \)

Example 2: Add \( (4y - 3) + (6y + 8) \)

\( 4y - 3 + 6y + 8 = 10y + 5 \)

Steps to Subtract Linear Expressions:

1. Distribute the negative sign (change all signs in the second expression)
2. Combine like terms
3. Simplify

Subtraction Examples:

Example 3: Subtract \( (5x + 7) - (2x + 3) \)

Distribute negative: \( 5x + 7 - 2x - 3 \)

Combine like terms: \( 3x + 4 \)

Example 4: Subtract \( (8a - 5) - (3a - 9) \)

Distribute negative: \( 8a - 5 - 3a + 9 \)

Combine: \( 5a + 4 \)

Mixed Operations:

Example 5: Simplify \( (7x + 4) - (3x - 2) + (x + 5) \)

\( 7x + 4 - 3x + 2 + x + 5 \)

\( = 5x + 11 \)

Key Formula:

\( (ax + b) + (cx + d) = (a + c)x + (b + d) \)

\( (ax + b) - (cx + d) = (a - c)x + (b - d) \)

6. Factors of Linear Expressions

Definition: Factoring is the reverse of the distributive property. It means rewriting an expression as a product.

Greatest Common Factor (GCF) Method:

Find the largest factor that divides evenly into all terms.

Steps to Factor:

1. Find the GCF of all coefficients
2. Find the GCF of all variables (lowest power)
3. Factor out the GCF
4. Write remaining terms in parentheses

Formula:

\( ab + ac = a(b + c) \)

Examples:

Example 1: Factor \( 6x + 9 \)

GCF of 6 and 9 is 3

\( 6x + 9 = 3(2x + 3) \)

Example 2: Factor \( 12y - 8 \)

GCF of 12 and 8 is 4

\( 12y - 8 = 4(3y - 2) \)

Example 3: Factor \( 5x^2 + 10x \)

GCF: 5 (from coefficients) and \( x \) (from variables)

\( 5x^2 + 10x = 5x(x + 2) \)

Example 4: Factor \( 18a^2 - 12a + 6 \)

GCF of 18, 12, and 6 is 6

\( 18a^2 - 12a + 6 = 6(3a^2 - 2a + 1) \)

Example 5: Factor \( -4x - 12 \)

GCF is -4 (factor out negative to make leading coefficient positive)

\( -4x - 12 = -4(x + 3) \)

Check Your Work:

Distribute to verify: \( 3(2x + 3) = 6x + 9 \) ✓

7 & 8. Identify Equivalent Linear Expressions (I & II)

Definition: Two expressions are equivalent if they have the same value for all values of the variable.

Method 1: Simplify Both Expressions

If both expressions simplify to the same form, they are equivalent.

Example: Are \( 3(x + 2) \) and \( 3x + 6 \) equivalent?

Simplify first: \( 3(x + 2) = 3x + 6 \)

Yes, they are equivalent! ✓

Method 2: Substitute Values

Test with one or more values. If results match for all test values, expressions are likely equivalent.

Example: Are \( 4x + 8 \) and \( 2(2x + 4) \) equivalent?

Test with \( x = 3 \):

\( 4(3) + 8 = 12 + 8 = 20 \)

\( 2(2(3) + 4) = 2(6 + 4) = 2(10) = 20 \) ✓

Common Equivalent Forms:

Practice Problems:

1. Are \( 5(x - 2) \) and \( 5x - 10 \) equivalent?

\( 5(x - 2) = 5x - 10 \) ✓ Yes!

2. Are \( 3x + 7 \) and \( 7 + 3x \) equivalent?

Yes, by commutative property ✓

3. Are \( 4(2x + 3) - 2x \) and \( 6x + 12 \) equivalent?

\( 4(2x + 3) - 2x = 8x + 12 - 2x = 6x + 12 \) ✓ Yes!

4. Are \( 2x + 5 \) and \( 2(x + 5) \) equivalent?

\( 2(x + 5) = 2x + 10 \) ✗ No! Different!

9. Identify Equivalent Linear Expressions: Word Problems

Strategy: Write expressions from word problems, then simplify to see if they're equivalent.

Example 1: Shopping Scenario

Situation: Maria buys 3 notebooks at $x each and 2 pens at $2 each. John buys 2 notebooks at $x each, 1 pen at $2, and a folder for $x + 2. Do they spend the same amount?

Maria's cost: \( 3x + 2(2) = 3x + 4 \)

John's cost: \( 2x + 2 + (x + 2) = 3x + 4 \)

Conclusion: Yes, they spend the same amount! ✓

Example 2: Perimeter Problem

Situation: Rectangle A has length \( x + 5 \) and width \( x \). Rectangle B has length \( 2x + 3 \) and width \( \frac{x}{2} \). Do they have the same perimeter?

Rectangle A perimeter: \( 2(x + 5) + 2x = 2x + 10 + 2x = 4x + 10 \)

Rectangle B perimeter: \( 2(2x + 3) + 2(\frac{x}{2}) = 4x + 6 + x = 5x + 6 \)

Conclusion: No, different perimeters ✗

Example 3: Payment Plans

Plan A: Pay $50 upfront plus $30 per month

Plan B: Pay $20 per month for 3 months, then $40 per month

Are the costs equivalent after \( n \) months (where \( n > 3 \))?

Plan A cost: \( 50 + 30n \)

Plan B cost: \( 20(3) + 40(n - 3) = 60 + 40n - 120 = 40n - 60 \)

Conclusion: Not equivalent ✗

Example 4: Age Problem

Situation: Sarah is \( x \) years old. Her brother is 3 years younger. In 5 years, will Sarah be twice as old as her brother is now?

Brother's age now: \( x - 3 \)

Sarah's age in 5 years: \( x + 5 \)

Twice brother's current age: \( 2(x - 3) = 2x - 6 \)

Compare: \( x + 5 \) vs. \( 2x - 6 \) — Not equivalent for all \( x \) ✗

Key Tips for Word Problems:

• Write an expression for each situation
• Simplify both expressions completely
• Compare the simplified forms
• Test with a specific value if unsure

Quick Reference: Properties and Operations

💡 Key Tips for Equivalent Expressions

• ✓ Use properties correctly: Know which operations are commutative and associative
• ✓ Distribute carefully: Multiply each term inside parentheses by the outside term
• ✓ Watch negative signs: When subtracting, change all signs in the second expression
• ✓ Combine only like terms: Same variables with same exponents
• ✓ Factor out GCF completely: Find the largest common factor
• ✓ Check equivalence: Simplify both expressions or test with values
• ✓ Show all steps: Don't skip steps when simplifying
• ✓ Verify by distributing: After factoring, distribute to check your answer
• ✓ Order matters for subtraction: \( a - b \neq b - a \)
• ✓ Read word problems carefully: Translate situations into expressions accurately