Algebraic Expressions
Learn to identify variables, coefficients, and constants, collect like terms, and substitute values into expressions.
What is an Algebraic Expression?
An algebraic expression is a mathematical phrase that uses numbers, letters (called variables), and operations. Unlike an equation, it does not have an equals sign.
Parts of an Expression
This expression has three terms: 3x, 2y, and 5.
Bar Model: What does 3x + 2 look like?
If x = 4, then 3x + 2 means "three groups of x, plus 2 more":
Total: 4 + 4 + 4 + 2 = 14
Collecting Like Terms
Like terms have the same variable raised to the same power. You can add or subtract like terms to simplify an expression.
Like Terms (CAN combine)
- 3x and 5x → 8x
- −2y and 7y → 5y
- 4 and −1 → 3
Unlike Terms (CANNOT combine)
- 3x and 2y — different variables
- 5x and 5x² — different powers
- 4a and 7 — one has a variable, one does not
Example: Simplify 4x + 3y − 2x + 5y + 1
Group like terms: (4x − 2x) + (3y + 5y) + 1
Result: 2x + 8y + 1
Key Vocabulary
Variable
A letter that represents an unknown or changing value. Examples: x, y, n.
Coefficient
The number multiplied by a variable. In 5x, the coefficient is 5.
Constant
A fixed number in an expression that has no variable attached. Example: the 7 in 3x + 7.
Term
A single part of an expression, separated by + or − signs. In 2x + 3y − 1, there are three terms.
Like Terms
Terms with the same variable and power. 4x and −2x are like terms.
Substitution
Replacing a variable with a number to calculate the value of an expression.
Worked Examples
Simplify: 7a + 3b − 2a + b
Step 1: Identify like terms. The "a" terms: 7a and −2a. The "b" terms: 3b and b.
Step 2: Collect the "a" terms: 7a − 2a = 5a
Step 3: Collect the "b" terms: 3b + b = 4b (remember: b = 1b)
Answer: 5a + 4b
Evaluate 2x + 3y when x = 4 and y = −1
Step 1: Substitute x = 4: 2(4) = 8
Step 2: Substitute y = −1: 3(−1) = −3
Step 3: Add the results: 8 + (−3) = 8 − 3 = 5
Answer: 5
Write an expression: "Five more than twice a number n"
Step 1: "Twice a number n" means 2n
Step 2: "Five more than" means + 5
Answer: 2n + 5
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Simplify: 6x + 2x − 3
Question 2
What is the coefficient of y in the expression 4x − 7y + 2?
Question 3
Evaluate 3a − 2b when a = 5 and b = 3.
Question 4
Simplify: 5m + 2n − 3m + 4n − 1
Question 5
Which expression represents "three less than four times a number p"?
Key Concepts Summary
- ● An algebraic expression contains variables, coefficients, constants, and operations.
- ● A variable is a letter (like x or y), a coefficient is the number in front, and a constant is a plain number.
- ● Like terms share the same variable and power and can be combined by adding/subtracting their coefficients.
- ● Substitution means replacing a variable with a given number and calculating the result.
- ● When writing expressions from words, translate carefully: "more than" means +, "less than" means −, "times" means ×.