Intro Algebra: Variables & Expressions

Variables are letters that stand for unknown numbers, and expressions combine those letters with numbers and operations to describe math situations.

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Definition

In algebra, a variable is a letter (like x, n, or t) used to represent a number we do not know yet or a number that can change. An expression is a math phrase made up of numbers, variables, and operations (like addition, subtraction, multiplication, or division) but with NO equals sign. For example, 3x + 5 is an expression — it has a variable (x), a number (5), and operations (multiplication and addition).

Remember the rule

To evaluate any expression: SUBSTITUTE the given number for the variable, then SIMPLIFY using order of operations (PEMDAS).

Key words

Variable: A letter that stands in for an unknown or changing number, like x, y, or n.
Expression: A math phrase with numbers, variables, and operations but no equals sign — for example, 4n + 2.
Coefficient: The number multiplied by a variable. In 7x, the coefficient is 7.
Constant: A plain number in an expression that never changes, like the 9 in 2x + 9.
Term: Each separate part of an expression separated by + or −. In 3x + 5, the terms are 3x and 5.
Evaluate: Plug a number in for the variable and calculate the answer.
Substitution: Replacing a variable with a specific number so you can evaluate the expression.
Like Terms: Terms that have the same variable (and same exponent), like 3x and 5x — they can be combined.

Worked examples

Write an expression: A bag has some apples. You add 4 more. How many apples are there?

→ n + 4 · We use n for the unknown starting number of apples, then add 4.

Identify the parts of 6x + 3: What is the coefficient, the variable, and the constant?

→ Coefficient: 6 | Variable: x | Constant: 3 · 6x means 6 times x; the 3 is on its own with no variable, so it is the constant.

Evaluate 5x + 2 when x = 3.

→ 5(3) + 2 = 15 + 2 = 17 · Replace x with 3, multiply first (order of operations), then add.

Evaluate 20 − 4n when n = 2.

→ 20 − 4(2) = 20 − 8 = 12 · Substitute 2 for n, handle multiplication before subtraction.

Simplify by combining like terms: 3x + 7x.

→ 10x · 3x and 7x both have the variable x, so add the coefficients: 3 + 7 = 10.

Write an expression: Tickets cost $8 each. How much do t tickets cost?

→ 8t · 8t means 8 times t; writing a number right next to a variable means multiplication.

Common mistakes

Forgetting that a number written next to a variable means multiplication — 6x means 6 TIMES x, not 6 and x added together.
Trying to solve an expression as if it were an equation — expressions do not have an equals sign and cannot be 'solved,' only evaluated or simplified.
Skipping order of operations when evaluating — always multiply/divide before adding/subtracting.
Combining terms that are NOT alike — for example, adding 3x + 5 and writing 8x is wrong because 3x and 5 are not like terms.
Dropping the variable by accident — writing 5x + 3x = 8 instead of 8x.

FAQs

Why do we use letters in math? Can we use any letter?

Letters are used because we do not know the exact number yet, or the number can change. You can use almost any letter, but x, y, n, and t are very common. The letter you choose does not change the math.

What is the difference between an expression and an equation?

An expression is a math phrase with no equals sign, like 3x + 1. An equation has an equals sign and shows two things are equal, like 3x + 1 = 10. You solve equations; you evaluate or simplify expressions.

What does 4n mean — is there a multiplication sign hiding there?

Yes! Writing a number directly next to a variable (like 4n) is a shorthand for 4 × n. Mathematicians drop the × sign to keep things neater and to avoid confusing it with the letter x.

Can an expression have more than one variable?

Absolutely. An expression like 3x + 2y has two variables. You would need the value of both x and y to evaluate it fully.

How do I know which terms are 'like terms'?

Like terms must have exactly the same variable part. So 5x and 2x are like terms (both have x), but 5x and 2y are NOT like terms because the variables are different.

My child's teacher writes 'the product of 6 and a number n' — what does that mean as an expression?

'Product' means multiplication, so 'the product of 6 and a number n' translates to the expression 6n.

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