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\(x\)
\(+ 5\)

Writing Algebraic
Expressions

Learn to translate words into math by identifying variables, constants, and operations to build algebraic expressions.

Pre-Algebra

What Is an Algebraic Expression?

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Unlike an equation, it does not have an equals sign.

\(3\)
\(x\)
\(+\)
\(7\)
 coefficient variable operation constant Expression:
\(3x + 7\)

Key Vocabulary

Variable
A letter (like \(x\), \(n\), or \(y\)) that represents an unknown value that can change.
Constant
A fixed number that does not change, such as \(5\), \(-3\), or \(\frac{1}{2}\).
Coefficient
The number multiplied by a variable: in \(4x\), the coefficient is \(4\).

Why Expressions Matter

Algebraic expressions let you model real-world situations with math. Instead of writing "five more than an unknown number," you can write \(x + 5\) and then use it to solve problems.

Shopping
"Total cost for \(n\) items at $3 each" becomes \(3n\).
Sports
"7 more points than the opponent's score \(s\)" becomes \(s + 7\).

The Translation Rule

To write an algebraic expression from words, follow this core principle:

Verbal Description Translate Algebraic Expression

Operation Keywords

Operation Key Words Example
Addition (\(+\)) sum, plus, more than, increased by, total \(x + 4\)
Subtraction (\(−\)) difference, minus, less than, decreased by, fewer \(x - 3\)
Multiplication (\(×\)) product, times, of, each, per, twice, double \(5x\)
Division (\(÷\)) quotient, divided by, per, ratio, split \(\frac{x}{2}\)

Expression vs. Equation

Expression
A mathematical phrase with no equals sign.
Example: \(2x + 7\)
It can be evaluated but not solved.
Equation
A mathematical sentence with an equals sign.
Example: \(2x + 7 = 15\)
It can be solved for the variable.

How to Translate Step by Step

1
Read carefully. Identify what quantity is unknown and assign it a variable.
2
Spot the keyword. Determine which operation the words describe (sum, product, difference, quotient).
3
Watch the order. "Less than" and "subtracted from" reverse the order of the terms.
4
Write the expression. Combine the variable, coefficient, operation, and constant into one expression.
5
Check by substitution. Plug in a simple number for the variable and see if the result makes sense.

Example 1: Addition

Eight more than a number
\(8\)
\(+\)
\(n\)

Example 2: Subtraction Order

"twice a number"
\(2x\)
 "five less than"
\(− 5\)
\(2x - 5\)
 "less than" reverses: subtract FROM the first part

Example 3: Multi-Step Phrase

1
Unknown: "a number" → let it be \(x\)
2
"Three more than a number" → \(x + 3\)
3
"The quotient of ... and four" → divide the first part by \(4\)
4
Expression: \(\displaystyle \frac{x + 3}{4}\)

Quick-Fire Translations

Verbal Phrase Expression
The sum of a number and 12 \(x + 12\)
Seven times a number \(7x\)
A number decreased by 9 \(x - 9\)
The product of 4 and a number, plus 1 \(4x + 1\)
Half of a number \(\frac{x}{2}\)
10 less than triple a number \(3x - 10\)

Choosing Meaningful Variables

While any letter works as a variable, choosing a letter that matches the context makes your expression easier to read and understand.

\(t\) for time
"Distance after \(t\) hours at 60 mph" → \(60t\)
\(p\) for price
"Price after a $5 discount" → \(p - 5\)
\(n\) for number
"Four more than a number" → \(n + 4\)

Multiplication Conventions

In algebra, we drop the multiplication sign and follow specific writing conventions to keep expressions clean and unambiguous.

1
Number before variable: Write \(5x\), not \(x5\). The coefficient always comes first.
2
No multiplication dot with variables: Write \(3y\), not \(3 \cdot y\) or \(3 \times y\).
3
Coefficient of 1 is hidden: Write \(x\), not \(1x\). Same idea: \(-x\) means \(-1 \cdot x\).

Common Mistakes

✕ Reversed Subtraction
Writing "3 less than \(x\)" as \(3 - x\) instead of the correct \(x - 3\). "Less than" means subtract from the other quantity.
✕ Adding an Equals Sign
Writing \(x + 5 = \text{?}\) when asked for an expression. An expression has no equals sign.
✕ Ignoring "Each" or "Per"
Translating "$3 per item for \(n\) items" as \(3 + n\) instead of the correct \(3n\). "Per" and "each" signal multiplication.

Summary Cheat Sheet

Component What It Is Example in \(3x + 7\)
Variable Unknown quantity \(x\)
Coefficient Number times the variable \(3\)
Constant Fixed value \(7\)
Operation Action between terms \(+\)
Term Each part separated by + or − \(3x\) and \(7\)

Question 1 of 5

Which expression represents "nine more than a number"?

Question 2 of 5

Translate: "Six less than twice a number"

Question 3 of 5

In the expression \(4y + 11\), what is the coefficient?

Question 4 of 5

Which verbal phrase matches the expression \(\frac{x + 5}{3}\)?

Question 5 of 5

A movie ticket costs \(\$8\). Which expression gives the total cost for \(n\) tickets plus a \(\$5\) service fee?