Writing Algebraic Expressions

Learn how to translate words and real-world situations into algebraic expressions using variables and operations.

Intermediate25 minLesson

Definition

Writing algebraic expressions means translating words into math using numbers, variables, and operations.

A variable (like

x

n

, or

y

) represents an unknown number.

Key Phrases to Know

Operation	Key Words	Example Phrase	Expression
Addition ( $+$ )	sum, plus, more than, increased by, total	five more than a number	$x + 5$
Subtraction ( $-$ )	difference, minus, less than, decreased by, fewer	a number decreased by 3	$x - 3$
Multiplication ( $\times$ )	product, times, of, twice, triple	twice a number	$2x$
Division ( $\div$ )	quotient, divided by, ratio, per	a number divided by 4	$\frac{x}{4}$

Important: In "less than" and "more than" phrases, the order matters!

"5 more than $x$ " means $x + 5$
"5 less than $x$ " means $x - 5$ (not $5 - x$ )

Try it now

Which expression represents "the sum of a number and 8"?

Worked Examples

Write an expression for: "The sum of a number and 12"

Identify the operation

"Sum" means addition ( $+$ ) → Operation: $+$

Identify the quantities

"A number" (unknown) and 12 → Variable and 12

Choose a variable

Let $n$ represent "a number" → $n$

Write the expression

Sum means add: $n + 12$

Answer: $n + 12$

Common Mistakes

Writing "5 less than $x$ " as $5 - x$ instead of $x - 5$

Why it's wrong: "Less than" reverses the order! "5 less than $x$ " means start with $x$ and take away 5.

Correct: "5 less than $x$ " = $x - 5$ . Think: "I have $x$ dollars and spend 5" leaves $x - 5$ .

Forgetting to use parentheses when needed

Why it's wrong: "Twice the sum of $x$ and 3" needs parentheses to show you add first.

Correct: "Twice the sum of $x$ and 3" = $2(x + 3)$ , not $2x + 3$

Confusing "a number" with a specific number

Why it's wrong: "A number" means an unknown value that should be a variable.

Correct: Always use a variable ( $x$ , $n$ , $y$ ) when you see "a number," "some number," or "an unknown."

Interactive Visual

Balance Scale

x + 3=7

Apply to both sides:

Solution: x = 4

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems

Problem 1 of 16

Easy

Which expression represents "the sum of a number and 8"?

Why It Matters

Writing algebraic expressions is the foundation of algebra! Here's why it matters:

Problem Solving: Real-world problems are written in words. To solve them, you must first translate them into mathematical expressions.

Communication: Expressions let you describe patterns and relationships concisely. Instead of saying "add 5 to whatever number you have," you write $x + 5$ .

Programming: Every app, game, and website uses expressions. When a game calculates your score as "base points times bonus multiplier," that's $p \times m$ !

Science: Formulas like distance = speed $\times$ time ( $d = st$ ) are algebraic expressions describing how the world works.

Real World Applications

Shopping and Budgeting

Stores use expressions to calculate totals with discounts, tax, and multiple items.

Example:

If shirts cost 15 dollars each and you have a 10 dollar coupon, the cost for $s$ shirts is $15s - 10$ .

1Try It Yourself

A streaming service charges 8 dollars per month plus a one-time 20 dollar setup fee.

Write an expression for the total cost after $m$ months.

Step 1: Write the mathematical expression

Monthly cost times months plus setup fee:

Sports Statistics

Athletes and coaches use expressions to track performance and predict outcomes.

Example:

If a basketball player scores $p$ points per game, their total after $g$ games is $pg$ .

2Try It Yourself

A runner improves by 30 seconds each week. Their initial time was 8 minutes (480 seconds).

Write an expression for their time after $w$ weeks of training.

Step 1: Write the mathematical expression

Initial time minus improvement per week times weeks:

Cooking and Recipes

Recipes are scaled using expressions when cooking for different numbers of people.

Example:

A recipe uses 2 cups of flour per person. For $p$ people, you need $2p$ cups.

3Try It Yourself

A pizza recipe needs 3 cups of cheese for every 2 pizzas, plus 1 extra cup for the crust.

Write an expression for the cheese needed for $n$ pizzas (where $n$ is even).

Step 1: Write the mathematical expression

Cheese per 2 pizzas times number of pairs plus extra:

Key Takeaways

1Algebraic expressions translate words into math using variables and operations
2Key words signal operations: sum (+), difference (-), product (x), quotient (/)
3Watch word order carefully: "5 less than $x$ " means $x - 5$ , not $5 - x$
4Use parentheses to group operations: "twice the sum" = $2(x + y)$
5Variables represent unknown or changing quantities in real situations

Frequently Asked Questions

No! You can use any letter. However, some letters are traditional:

t

for time,

d

for distance,

n

for "a number." Use whatever makes sense for the problem.

Glossary

Algebraic expression: A combination of numbers, variables, and operations (like $3x + 5$ )
Variable: A letter that represents an unknown or changing number (like $x$ , $n$ , or $t$ )
Coefficient: The number multiplied by a variable (in $3x$ , the coefficient is 3)
Constant: A number without a variable (in $2x + 5$ , the constant is 5)
Term: A single part of an expression separated by + or - (in $3x + 5$ , the terms are $3x$ and $5$ )

Writing Algebraic Expressions

Definition

Key Phrases to Know

Worked Examples

Common Mistakes

Interactive Visual

Balance Scale

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Shopping and Budgeting

Sports Statistics

Cooking and Recipes

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Variables and Algebraic Expressions

Simplifying Algebraic Expressions

Evaluating Expressions

Combining Like Terms