Variables and Algebraic Expressions

Learn what variables are and how to write, read, and evaluate algebraic expressions.

Intermediate25 minLesson
Printable worksheet

Definition

A variable is a letter (like xx, yy, or nn) that represents an unknown number or a number that can change.
An algebraic expression is a mathematical phrase that combines:
  • Numbers (constants like 55, 1212, 12\frac{1}{2})
  • Variables (letters like xx, yy, aa)
  • Operations (like ++, −-, ×\times, ÷\div)
Examples of algebraic expressions:
3x+53x + 5
2y−72y - 7
n4+10\frac{n}{4} + 10
Unlike equations, expressions do not have an equals sign.

Try it now

Which of the following is a variable?

Worked Examples

Write an algebraic expression for: "five more than a number"
1

Choose a variable for the unknown number

Let nn represent the unknown number → Variable: nn
2

Identify the operation

"More than" means addition → Operation: ++
3

Write the expression

Five more than nn means n+5n + 5

Common Mistakes

Writing 3x3x as 3+x3 + x instead of 3×x3 \times x

Why it's wrong: In algebra, when a number is written next to a variable without a sign, it means multiplication. 3x3x means "3 times xx", not "3 plus xx".

Correct: 3x=3×x3x = 3 \times x. So if x=5x = 5, then 3x=153x = 15 (not 88).

Forgetting order of operations when evaluating

Why it's wrong: When substituting values, you must still follow PEMDAS/BODMAS. Multiplication comes before addition.

Correct: For 2+3x2 + 3x with x=4x = 4: First 3×4=123 \times 4 = 12, then 2+12=142 + 12 = 14.

Using the same variable for different unknowns

Why it's wrong: If you have two different unknown quantities, they need different variables.

Correct: If age and height are both unknown, use aa for age and hh for height, not xx for both.

Interactive Visual

Algebra tiles: combining like terms

Each tile is a term: big squares are x², long strips are x, small squares are 1. Red tiles are negative.

3x3x
77

3x+73x + 7

Every term becomes tiles: the coefficient tells how many.

=x2= x^2=x= x=1= 1negative

Function graph

-10123456051015202530xy(4, 19)

Tap or hover over the graph to read off values.

3x + 7

Table of values

x01234
y710131619

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which of the following is a variable?

Why It Matters

Variables and expressions are the foundation of algebra and appear everywhere:
  • Formulas: The area of a rectangle is A=l×wA = l \times w (length times width)
  • Real life: If a movie ticket costs 12 dollars, the cost for nn people is 12n12n dollars
  • Science: Distance equals speed times time: d=std = st
  • Computer programming: Variables store changing values in every program
Once you understand variables, you can describe patterns, solve problems, and create formulas that work for ANY numbers!

Real World Applications

Shopping and Discounts

Stores use expressions to calculate prices. If jeans cost 40 dollars and shirts cost $s$ dollars each, buying 2 shirts with jeans costs $40 + 2s$ dollars.

Example:

If shirts are 25 dollars each: 40+2(25)=40+50=9040 + 2(25) = 40 + 50 = 90 dollars total.

1Try It Yourself

A pizza shop charges 12 dollars for a pizza plus 2 dollars per topping.

Write an expression for the cost with tt toppings, then find the cost of a pizza with 5 toppings.

Step 1: Write the mathematical expression

Cost = base price + (price per topping × number of toppings):

Distance and Travel

If you drive at $s$ kilometers per hour for $t$ hours, you travel $s \times t$ kilometers.

Example:

Driving at 60 km/h for 3 hours: 60×3=18060 \times 3 = 180 km.

2Try It Yourself

A train travels at 80 kilometers per hour.

Write an expression for the distance after hh hours, then find how far it goes in 4 hours.

Step 1: Write the mathematical expression

Distance = speed × time:

Savings and Money

Expressions help track money over time. If you have 50 dollars and save $w$ dollars each week, after one week you have $50 + w$ dollars.

Example:

Saving 15 dollars per week: after 1 week you have 50+15=6550 + 15 = 65 dollars.

3Try It Yourself

You have 100 dollars and spend 8 dollars each day on lunch.

Write an expression for the money left after dd days, then find how much is left after 5 days.

Step 1: Write the mathematical expression

Money left = starting amount - (daily spending × days):

Key Takeaways

  • 1A variable is a letter representing an unknown or changing number
  • 2An algebraic expression combines numbers, variables, and operations (no equals sign)
  • 3When a number is next to a variable (like 3x3x), it means multiplication: 3×x3 \times x
  • 4To evaluate an expression, substitute values for variables and calculate
  • 5Always follow order of operations (PEMDAS) when evaluating

Frequently Asked Questions

Letters are clearer and allow us to write rules. For example, A=lwA = lw tells us exactly how to find area, while "Area = ___ times ___" is ambiguous.
Letters are clearer and allow us to write rules. For example, A=lwA = lw tells us exactly how to find area, while "Area = ___ times ___" is ambiguous.
Usually no, but some letters have common meanings: xx for unknowns, tt for time, dd for distance, nn for number of items. Choose letters that make sense for your problem.
An expression is a mathematical phrase without an equals sign (like 3x+53x + 5). An equation has an equals sign and shows two things are equal (like 3x+5=143x + 5 = 14).

Glossary

Variable
A letter that represents an unknown or changing number (e.g., xx, nn, tt)
Algebraic expression
A combination of numbers, variables, and operations without an equals sign
Constant
A number that does not change (e.g., 55, −3-3, 12\frac{1}{2})
Coefficient
The number multiplied by a variable (in 7x7x, the coefficient is 77)
Term
A single part of an expression separated by ++ or −- (in 3x+53x + 5, the terms are 3x3x and 55)
Evaluate
To find the value of an expression by substituting numbers for variables
Today's challengeOne problem, three tries

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