What is an Algebraic Expression?

An algebraic expression is a mathematical phrase made up of:

• Numbers
• Variables (letters)
• Mathematical operations

Unlike an equation, an expression does not contain an equals sign (=).

Examples include:

$x+5$x+5$3y$3y$4a-7$4a−7$2m+3n$2m+3n

Algebraic expressions describe quantities and relationships.

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Parts of an Algebraic Expression

Every algebraic expression is made up of different parts.

Consider the expression:

$5x+3$5x+3

It contains:

• Term:$5x$5x
• Term:$3$3
• Coefficient: 5
• Variable:$x$x
• Constant: 3

Understanding these parts makes expressions easier to work with.

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Terms

A term is a single part of an expression separated by addition (+) or subtraction (−) signs.

Example:

$4x+7-2y$4x+7−2y

The three terms are:

• $4x$4x
• $7$7
• $-2y$−2y

Each term is treated as one unit.

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Variables

A variable is a letter that represents a number.

Examples include:

• $x$x
• $y$y
• $a$a
• $n$n

Variables may represent:

• Unknown numbers.
• Quantities that can change.

Example:

$6x$6x

The variable is:

$x$x

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Coefficients

A coefficient is the number multiplied by a variable.

Example:

$8y$8y

The coefficient is:

$8$8

Another example:

$-3a$−3a

The coefficient is:

$-3$−3

If no number is written, the coefficient is 1.

Example:

$x=1x$x=1x

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Constants

A constant is a number without a variable.

Examples:

• 7
• −5
• 12

Constants have fixed values.

Example:

$4x+9$4x+9

The constant is:

$9$9

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Identifying the Parts

Example:

$3a-5+2b$3a−5+2b

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Writing Algebraic Expressions

Words can be translated into algebraic expressions.

Examples:

A number plus 6

$x+6$x+6

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Five more than a number

$x+5$x+5

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Seven times a number

$7n$7n

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A number divided by 4

$\frac{x}{4}$4x​

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Three less than a number

$x-3$x−3

Carefully reading mathematical language helps produce the correct expression.

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Translating Mathematical Phrases

Certain words represent specific mathematical operations.

Learning these words makes it easier to write algebraic expressions.

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Be Careful with "Less Than"

The phrase less than reverses the order.

Example:

5 less than a number

means:

$x-5$x−5

However:

5 is less than a number

means:

$x>5$x>5

And:

5 less than 20

means:

$20-5$20−5

Always read the phrase carefully to determine the correct order.

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Expressions vs Equations

Students often confuse expressions and equations.

Expression

Contains:

• Numbers.
• Variables.
• Operations.

No equals sign.

Example:

$4x+2$4x+2

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Equation

Contains:

• Two expressions.
• An equals sign.

Example:

$4x+2=18$4x+2=18

Expressions describe quantities.

Equations state that two quantities are equal.

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Interpreting Expressions

Expressions often describe real situations.

Example:

$5n$5n

If:

$n$

n represents the number of notebooks,

then:

$5n$5n

means:

"The total cost if each notebook costs $5."

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Another example:

$2a+6$2a+6

If:

$a$

a represents age,

then:

$2a+6$2a+6

means:

"Twice the person's age plus six."

The meaning depends on what the variable represents.

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Real-World Models

Expressions help model many situations.

Shopping

Each apple costs $2.

Buying

$a$

a apples costs:

$2a$2a

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Taxi Fare

Starting fee:

$4

Plus:

$3 per kilometre.

Expression:

$4+3k$4+3k

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Perimeter

Rectangle:

Length =

$l$

l

Width =

$w$

w

Perimeter:

$2l+2w$2l+2w

Expressions make mathematical models concise and flexible.

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Why Algebraic Expressions are Important

Expressions allow mathematicians and scientists to:

• Describe patterns.
• Write formulas.
• Solve problems.
• Model real situations.
• Communicate mathematical ideas efficiently.

They are one of the building blocks of algebra.

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Worked Example

Write an algebraic expression for:

"Eight more than three times a number."

Solution

Step 1:

Let the number be:

$x$x

Step 2:

Three times the number:

$3x$3x

Step 3:

Add eight:

$3x+8$3x+8

The expression is:

$\boxed{3x+8}$3x+8​

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Key Terms

Algebraic Expression — A mathematical phrase containing numbers, variables, and operations, but no equals sign.

Term — A single part of an expression separated by addition or subtraction signs.

Variable — A letter representing a number.

Coefficient — The numerical factor multiplied by a variable.

Constant — A number without a variable.

Expression — A mathematical phrase without an equals sign.

Equation — A mathematical statement showing that two expressions are equal.

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Key Takeaways

• An algebraic expression contains numbers, variables, and operations but does not contain an equals sign.
• Expressions are made up of terms, coefficients, variables, and sometimes constants.
• Mathematical phrases can be translated into algebraic notation using the correct operations.
• Expressions describe quantities, while equations state that two quantities are equal.
• The meaning of an expression depends on what its variables represent.
• Algebraic expressions are used to model patterns, formulas, and real-world situations.

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Suggested Images

Suggested placement:

1. After "Parts of an Algebraic Expression" – Labelled diagram of an expression such as 5x + 3, identifying the terms, coefficient, variable, and constant.
2. After "Writing Algebraic Expressions" – Infographic translating common verbal phrases (e.g., "five more than a number") into algebraic notation.
3. After "Expressions vs Equations" – Side-by-side comparison showing an expression and an equation, highlighting the presence or absence of an equals sign.
4. After "Real-World Models" – Everyday examples such as shopping costs, taxi fares, and rectangle perimeter represented with algebraic expressions.
5. After "Translating Mathematical Phrases" – Reference chart linking words like sum, difference, product, and quotient to their corresponding mathematical operations.
6. Near "Why Algebraic Expressions are Important" – Illustration of a student using algebraic expressions to solve a real-world problem or identify a number pattern.