ALGEBRAIC EXPRESSION

Definition of Algebraic Expression

An algebraic expression is a mathematical phrase made from numbers, variables (like x or n), and operations such as addition, subtraction, multiplication, division, and exponents. It does not include an equals sign.

Why algebraic expressions matter

• They let you describe patterns and real-world situations with variables.
• They are the building blocks for equations, formulas, and functions.
• You use them in geometry, physics, finance, coding, and more.

Parts of an algebraic expression

Here’s what to look for inside an expression:

• Variable: a letter that stands for an unknown value (example: x, n).
• Coefficient: a number multiplying a variable (in 7x, the coefficient is 7).
• Term: a single piece separated by + or − signs (3x is a term, -5 is a term).
• Constant: a number with no variable (example: 12).
• Operator: symbols like +, −, ×, ÷, ^.

Examples of algebraic expressions

Like terms (a key idea)

Like terms have the same variable part (same letters with the same exponents). You can combine only like terms.

• 3x and 5x are like terms → 3x + 5x = 8x
• 2x and 2x² are not like terms.
• 7ab and -3ab are like terms → 4ab

Evaluating an expression

To evaluate an expression, substitute a value for the variable and compute.

Example: Evaluate 2x + 3 when x = 4.

2(4) + 3 = 8 + 3 = 11

Simplifying an expression

To simplify, combine like terms and use order of operations.

Example: Simplify 4x + 3 - 2x + 9.

Combine like terms: (4x - 2x) + (3 + 9) = 2x + 12

Common mistakes

• Combining unlike terms: x + x² cannot become 2x² (that’s wrong).
• Forgetting parentheses: 3(x - 2) means multiply both terms → 3x - 6.
• Confusing an expression with an equation: expressions have no equals sign.

Quick practice (with answers)

1. Simplify: 6y + 2 - 3y + 5
2. Evaluate: n² - 4 when n = 5
3. Are these like terms? 2ab and -7ba

Answers: 1) 3y + 7   2) 21   3) Yes (same letters)

Video example: Algebraic expressions

Solved examples

Question: A shop charges $5 for renting a video cassette for the first day. It charges $2 extra for every additional day. Which expression gives the total cost for n days?

Choices:

A. 3 + 2n
B. 5n + 2
C. n
D. 5 - 2n
Correct Answer: A

Solution

Day 1 costs $5. The remaining n-1 days cost $2 dollars each.
Total: 5 + 2(n - 1) = 5 + 2n - 2 = 3 + 2n

Example 2: Simplify an expression

Question: Simplify 4x - 3 + 2x + 5.
Solution: Combine like terms: 4x + 2x = 6x and -3 + 5 = 2.
Answer: 6x + 2

Example 3: Evaluate an expression

Question: Evaluate 3(a - 2) + 4 when a = 5.
Solution: Substitute a=5: 3(5-2)+4 = 3× 3 + 4 = 9 + 4.
Answer: 13

Example 4: Write an expression from words

Question: Write an algebraic expression for “twice a number decreased by 7”.
Solution: Let the number be n. “Twice a number” is 2n. “Decreased by 7” means subtract 7.
Answer: 2n - 7

Extra practice (with answers)

1. Simplify: 7y - 4 + 2y + 9
2. Evaluate: 2m² - 3m + 1 when m = -2
3. Write an expression: “three more than 5 times a number”
4. Are these like terms? 8a² and -3a

Answers: 1) 9y + 5   2) 15   3) 5n + 3   4) No