Algebraic Expressions - Seventh Grade
Variables, Terms, Coefficients, Writing & Evaluating
1. Understanding Algebraic Expressions
Definition
An algebraic expression is a combination of
constants, variables, and operations (+, −, ×, ÷)
• No equal sign (that makes it an equation)
• Example: 3x + 5, 2y² − 7, 4a + 2b
Key Components
Variable: A letter representing an unknown value (x, y, a, b)
Constant: A fixed number (5, -3, 0.5)
Coefficient: Number multiplied by a variable (in 5x, coefficient is 5)
Term: Parts separated by + or − signs
2. Identifying Terms and Coefficients
What is a Term?
A term is a single part of an expression
separated by + or − signs
Example: In 3x² + 5x − 7, there are 3 terms:
• Term 1: 3x²
• Term 2: 5x
• Term 3: −7
What is a Coefficient?
A coefficient is the numerical part of a term
that is multiplied by the variable
Examples:
• In 7x, coefficient is 7
• In −4y, coefficient is −4
• In x, coefficient is 1 (understood)
• In −z, coefficient is −1
Example: Identify Terms and Coefficients
Expression: 5x² − 3x + 8
Terms:
• 5x² (coefficient: 5)
• −3x (coefficient: −3)
• 8 (constant term, no coefficient)
Variable: x
Constant: 8
3. Factors of Variable Expressions
What is a Factor?
A factor is a number or variable that is
MULTIPLIED to make a term
Example: In 6xy, the factors are 6, x, and y
Examples of Factors
| Expression | Factors |
|---|---|
| 12x | 12 and x |
| 5ab | 5, a, and b |
| 3x²y | 3, x, x, and y |
| −8mn | −8, m, and n |
4. Writing Expressions: One Operation
Key Translation Words
| Operation | Key Words | Example |
|---|---|---|
| Addition (+) | sum, plus, more than, increased by, total | x + 5 |
| Subtraction (−) | difference, minus, less than, decreased by, subtract from | x − 3 |
| Multiplication (×) | product, times, multiplied by, of, twice | 4x |
| Division (÷) | quotient, divided by, per, ratio | x/2 |
Examples
• "A number plus 7" → x + 7
• "5 less than a number" → x − 5
• "Twice a number" → 2x
• "A number divided by 3" → x/3
• "The product of 6 and n" → 6n
5. Writing Expressions: Two or Three Operations
Important: Order Matters!
Addition & Multiplication: Order doesn't matter
• x + 5 = 5 + x
• 3x = x × 3
Subtraction & Division: Order DOES matter!
• x − 5 ≠ 5 − x
• x/5 ≠ 5/x
Examples
Example 1: "Three times a number plus 5"
Three times a number → 3x
plus 5 → + 5
Expression: 3x + 5
Example 2: "5 more than twice a number"
twice a number → 2x
5 more than → + 5
Expression: 2x + 5
Example 3: "The sum of a number and 4, divided by 2"
sum of a number and 4 → (x + 4)
divided by 2 → ÷ 2
Expression: (x + 4)/2
6. Writing Expressions: Word Problems
Steps to Solve
Step 1: Read the problem carefully
Step 2: Identify the unknown (choose a variable)
Step 3: Look for key words
Step 4: Write the expression
Real-World Examples
Problem 1: Sarah has $20 more than twice what Tom has. If Tom has x dollars, write an expression for Sarah's money.
Tom's money = x
Twice Tom's money = 2x
$20 more than = + 20
Sarah's money: 2x + 20
Problem 2: A rectangle's length is 3 inches more than its width. If the width is w, write an expression for the perimeter.
Width = w
Length = w + 3
Perimeter = 2(length + width)
Perimeter: 2(w + w + 3) = 2(2w + 3) = 4w + 6
7. Evaluating Expressions
What Does "Evaluate" Mean?
To evaluate means to find the VALUE of
an expression by substituting numbers for variables
Steps to Evaluate
Step 1: SUBSTITUTE the given value for the variable
Step 2: SIMPLIFY using order of operations (PEMDAS)
Step 3: Calculate the final answer
Example: Linear Expression
Evaluate: 3x + 7 when x = 4
Step 1: Substitute 4 for x
3(4) + 7
Step 2: Multiply first
12 + 7
Step 3: Add
19
Answer: 19
8. Evaluating Multi-Variable Expressions
Example
Evaluate: 2a + 3b when a = 5 and b = 2
Step 1: Substitute values
2(5) + 3(2)
Step 2: Multiply
10 + 6
Step 3: Add
16
Answer: 16
Another Example
Evaluate: x² + 2xy − y when x = 3 and y = 4
Substitute: (3)² + 2(3)(4) − 4
Exponents: 9 + 2(3)(4) − 4
Multiply: 9 + 24 − 4
Add/Subtract: 29
Answer: 29
9. Evaluating Absolute Value Expressions
What is Absolute Value?
Absolute value is the DISTANCE from zero
Always POSITIVE or ZERO
Symbol: | |
• |5| = 5
• |−5| = 5
Example
Evaluate: |2x − 5| when x = 1
Step 1: Substitute
|2(1) − 5|
Step 2: Simplify inside | |
|2 − 5| = |−3|
Step 3: Find absolute value
3
Answer: 3
10. Evaluating Nonlinear Expressions
What is Nonlinear?
Nonlinear expressions contain
exponents, squares, cubes, or products of variables
Examples: x², 3n³, xy, 2ab
Example 1: Squares
Evaluate: x² + 4x when x = 5
Substitute: (5)² + 4(5)
Square first: 25 + 4(5)
Multiply: 25 + 20
Add: 45
Answer: 45
Example 2: Cubes
Evaluate: 2n³ − 10 when n = 2
Substitute: 2(2)³ − 10
Cube: 2(8) − 10
Multiply: 16 − 10
Subtract: 6
Answer: 6
Quick Reference: Order of Operations (PEMDAS)
P - Parentheses ( )
E - Exponents x², x³
M - Multiplication ×
D - Division ÷
A - Addition +
S - Subtraction −
💡 Important Tips to Remember
✓ Variable: A letter representing an unknown value
✓ Coefficient: Number multiplying the variable
✓ Term: Parts separated by + or − signs
✓ Writing expressions: Translate key words carefully
✓ "More than" and "less than" require careful ordering
✓ Evaluating: Always substitute first, then simplify
✓ Use PEMDAS for order of operations
✓ Parentheses matter! (x + 3) × 2 ≠ x + 3 × 2
✓ Absolute value: Always positive or zero
✓ Watch for negative signs when substituting
🧠 Memory Tricks & Strategies
PEMDAS:
"Please Excuse My Dear Aunt Sally"
Key Words for Addition:
"Sum, Plus, More, Increased - addition is the deed!"
Key Words for Subtraction:
"Difference, Minus, Less - subtraction's the test!"
Key Words for Multiplication:
"Product, Times, Of - multiply enough!"
Key Words for Division:
"Quotient, Per, Divided by - division will fly!"
Evaluating:
"Sub and Solve - that's how problems evolve!"
Order Matters:
"5 less than x is NOT 5 − x, it's x − 5, that's the fix!"
Master Algebraic Expressions! 📝 🔢
Remember: Substitute carefully and follow PEMDAS!