By Thread Academy · 18 September 2026 · Mathematics
Before you can solve equations, you need to speak the language they are written in. That language is built from algebraic expressions: combinations of numbers, letters, and operations that describe quantities without yet claiming that anything equals anything else. This guide introduces the vocabulary of algebra, shows you how to tidy expressions up, and teaches you to expand brackets with confidence.
Expressions, terms, and variables
An expression is a combination of numbers, variables, and operation signs with no equals sign. The moment an equals sign appears, it becomes an equation instead. So 4x + 7y - 2 is an expression, while 4x + 7y - 2 = 10 is an equation.
A term is a single part of an expression, separated from the next part by a plus or minus sign. In the expression 4x + 7y - 2, the terms are 4x, 7y, and -2. Note that the minus sign belongs to the term that follows it, so the third term is negative 2, not positive 2.
A variable is a letter, such as x or y, that stands for a number we do not know yet or a quantity that can change. In a formula for the area of a rectangle, A = lw, the letters l and w are variables because the length and width can take different values.
Coefficients
The coefficient of a variable is the number multiplying it. In the term 5x, the coefficient of x is 5. In the term -3a, the coefficient of a is -3, with the negative sign included.
A term with no variable, such as the -2 in 4x + 7y - 2, is called the constant term. Its value never changes.
Worked example 1: naming the parts
Identify the terms, the variables, the coefficients, and the constant term in 5m^2 - 2m + 9.
Step 1: Split into terms at the plus and minus signs. The terms are 5m^2, -2m, and 9.
Step 2: Find the variables. The letter m appears in the first two terms, so m is the variable.
Step 3: Read off the coefficients, keeping the signs. The coefficient of m^2 is 5, and the coefficient of m is -2.
Step 4: The term with no variable is 9, so 9 is the constant term.
Like terms
Like terms are terms that contain exactly the same variables raised to exactly the same powers. Only like terms can be combined.
- 3x and -7x are like terms, because both contain x to the first power.
- 3x and 3x^2 are not like terms, because the powers of x differ.
- 4ab and -2ab are like terms, because both contain the same variable pair.
- 4ab and 4a are not like terms, because one contains b and the other does not.
A useful picture: think of x as apples and y as bananas. You can add 3 apples to 5 apples, but 3 apples plus 2 bananas cannot be merged into a single pile. Like terms are the same fruit.
Simplifying expressions
Simplifying means collecting like terms together to write the expression as neatly as possible. Group the like terms, add or subtract their coefficients, and keep the unlike terms separate.
Worked example 2: collecting like terms
Simplify 4a + 3b - 2a + 5b.
Step 1: Group the like terms. The a terms are 4a and -2a. The b terms are 3b and 5b.
Step 2: Combine the a terms: 4a - 2a = 2a.
Step 3: Combine the b terms: 3b + 5b = 8b.
Answer: 2a + 8b.
Worked example 3: with squared terms
Simplify 2x^2 + 5x - x^2 + 3.
Step 1: Group the like terms. The x^2 terms are 2x^2 and -x^2. The x term is 5x, and 3 is a constant.
Step 2: Combine the x^2 terms: 2x^2 - x^2 = x^2.
Step 3: Nothing combines with 5x or with 3, so they stay as they are.
Answer: x^2 + 5x + 3.
Notice that x^2 and x were not combined. They are different powers, so they are unlike terms, and merging them would be as wrong as adding apples to bananas.
Expanding a single bracket
Expanding a bracket means multiplying the term outside by every term inside, removing the bracket. This uses the distributive law, and the golden rule is that every term inside gets multiplied, with its sign.
Worked example 4: a positive multiplier
Expand 3(x + 4).
Step 1: Multiply x by 3 to get 3x.
Step 2: Multiply 4 by 3 to get 12.
Answer: 3x + 12.
Worked example 5: signs inside the bracket
Expand 5(2y - 3).
Step 1: Multiply 2y by 5 to get 10y.
Step 2: Multiply -3 by 5 to get -15.
Answer: 10y - 15.
Worked example 6: a negative multiplier
Expand -2(a + 5).
Step 1: Multiply a by -2 to get -2a.
Step 2: Multiply 5 by -2 to get -10. A negative times a positive is negative.
Answer: -2a - 10.
Negative multipliers are the most common source of errors in this topic. Multiply the signs deliberately on every term rather than rushing, and expansion becomes routine.
Putting it together
Simplifying and expanding often appear in the same question: expand the brackets first, then collect like terms.
Worked example 7: expand, then simplify
Simplify 2(x + 3) + 3(x - 1).
Step 1: Expand the first bracket: 2x + 6.
Step 2: Expand the second bracket: 3x - 3.
Step 3: Collect like terms: 2x + 3x = 5x, and 6 - 3 = 3.
Answer: 5x + 3.
Key takeaways
- An expression has no equals sign; terms are its parts separated by plus and minus signs; variables are letters standing for unknown or changing numbers.
- The coefficient is the number multiplying a variable, sign included; the constant term has no variable.
- Only like terms, with identical variables and powers, can be combined.
- Simplify by grouping like terms and adding or subtracting their coefficients.
- Expand a bracket by multiplying the outside term by every term inside, taking care with negative signs.
- In mixed questions, expand first and simplify second.