Algebraic Expressions

Algebra allows us to represent numbers and relationships using letters, numbers, and mathematical operations. Instead of working only with known values, we can use symbols to represent values that may change or may not yet be known.

An algebraic expression is a mathematical phrase made from numbers, variables, and operations. Learning how to read and write expressions is one of the foundations of algebra.

Learning Targets

• I can identify terms, coefficients, and variables in an expression.
• I can write algebraic expressions from verbal descriptions.
• I can translate words and phrases into algebraic notation.
• I can distinguish between expressions and equations.
• I can interpret the meaning of an algebraic expression.

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What Is an Algebraic Expression?

An algebraic expression is a combination of:

• Numbers
• Variables
• Mathematical operations

For example:

3x + 7

is an algebraic expression.

It contains:

• The number 3
• The variable x
• The number 7
• Addition and multiplication

The expression represents a value, but it does not state that this value is equal to something else.

Other examples include:

5y

2a + 9

4x − 3

6m + 2n

x² + 5x − 4

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Variables

A variable is a letter or symbol used to represent a number.

Common variables include:

x, y, a, b, m, n

For example:

x + 5

The variable is:

x

The value of x could change.

If:

x = 3

then:

x + 5 = 3 + 5 = 8

If:

x = 10

then:

x + 5 = 10 + 5 = 15

The expression stays the same, but its value changes depending on the value of the variable.

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Why Do We Use Variables?

Variables allow us to describe general relationships rather than individual calculations.

Suppose one movie ticket costs $12.

The cost of one ticket is:

$12

The cost of two tickets is:

2 × $12

But what if we do not know how many tickets will be purchased?

Let:

n = number of tickets

Then the total cost can be represented by:

12n

This expression works for any number of tickets.

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Terms

A term is a single part of an algebraic expression.

Terms are separated by addition or subtraction signs.

Consider:

4x + 7

The terms are:

4x

and:

7

So the expression contains two terms.

Consider:

3x + 2y − 8

The terms are:

3x

2y

−8

This expression contains three terms.

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Identifying Terms

Consider:

5a − 3b + 7c + 10

The terms are:

5a

−3b

7c

10

There are four terms.

It is useful to include the sign in front of a term.

For example, in:

8x − 5

the second term can be thought of as:

−5

rather than simply 5.

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Coefficients

A coefficient is the numerical factor multiplying a variable.

Consider:

6x

The coefficient of x is:

6

In:

−4y

the coefficient of y is:

−4

In:

3a + 7b

the coefficient of a is:

3

and the coefficient of b is:

7

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Hidden Coefficients

Sometimes a coefficient is not written.

Consider:

x

This really means:

1x

Therefore, the coefficient is:

1

Similarly:

−x

means:

−1x

Therefore, the coefficient is:

−1

These coefficients are usually not written because:

1 × x = x

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Constants

A constant is a number that appears by itself without a variable.

Consider:

5x + 9

The constant is:

9

In:

3x − 7

the constant term is:

−7

In:

4a + 2b + 12

the constant is:

12

A constant has a fixed value, while a variable can represent different values.

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Putting the Parts Together

Consider the expression:

5x + 3y − 8

We can identify its parts.

Variables:

x and y

Coefficients:

5 and 3

Constant:

−8

Terms:

5x, 3y, and −8

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What Does 5x Mean?

In algebra, multiplication signs are often omitted.

Therefore:

5x

means:

5 × x

Similarly:

3a = 3 × a

and:

xy = x × y

We normally write:

4x

rather than:

4 × x

because the shorter form is easier to read and is standard algebraic notation.

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Writing Expressions from Words

One of the most important algebra skills is translating between words and mathematical notation.

For example:

Five more than x

becomes:

x + 5

The phrase describes starting with x and adding 5.

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Addition Phrases

Words that often indicate addition include:

• Plus
• Add
• Sum
• Increased by
• More than
• Total

Examples:

x plus 4

becomes:

x + 4

7 more than n

becomes:

n + 7

The sum of a and 10

becomes:

a + 10

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Subtraction Phrases

Words that often indicate subtraction include:

• Minus
• Subtract
• Difference
• Decreased by
• Less than
• Fewer than

Examples:

x minus 6

becomes:

x − 6

8 less than n

becomes:

n − 8

The difference between x and 5

can be written:

x − 5

Order is especially important when translating subtraction.

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Be Careful with "Less Than"

Consider:

5 less than x

This means:

x − 5

It does not mean:

5 − x

The words appear as:

5 less than x

but mathematically we start with x and remove 5.

Another example:

3 less than y

becomes:

y − 3

This is a very common source of errors.

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Multiplication Phrases

Words that often indicate multiplication include:

• Times
• Product
• Multiplied by
• Twice
• Double
• Triple
• Of

Examples:

Five times x

becomes:

5x

Twice n

becomes:

2n

The product of 7 and y

becomes:

7y

Triple a number x

becomes:

3x

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Division Phrases

Words that often indicate division include:

• Divided by
• Quotient
• Per
• Ratio

For example:

x divided by 4

can be written:

x / 4

The quotient of y and 5

can be written:

y / 5

Half of x

can be written:

x / 2

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Combining Operations

Many verbal descriptions contain more than one operation.

Consider:

Three times a number plus five

Let the number be x.

Three times the number is:

3x

Then add five:

3x + 5

Therefore:

Three times a number plus five → 3x + 5

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Worked Example — Translating Words

Translate:

Four more than twice a number

Let the number be x.

Twice the number:

2x

Four more:

2x + 4

Therefore:

Four more than twice a number → 2x + 4

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Worked Example — Another Translation

Translate:

Seven less than three times a number

Let the number be n.

Three times the number:

3n

Seven less than this:

3n − 7

Therefore:

Seven less than three times a number → 3n − 7

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Expressions with Grouping

Sometimes brackets are necessary to show which operation should happen first.

Consider:

Three times the sum of x and 4

The sum of x and 4 is:

x + 4

We need three times the entire sum.

Therefore:

3(x + 4)

This is different from:

3x + 4

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Worked Example — Using Brackets

Translate:

Five times the difference between x and 2

The difference is:

x − 2

Five times this entire difference is:

5(x − 2)

Without the brackets:

5x − 2

would represent something different.

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Expressions and Equations

An expression and an equation are not the same thing.

An expression represents a mathematical value.

For example:

3x + 5

An equation states that two expressions are equal.

For example:

3x + 5 = 20

The key difference is the equals sign.

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Comparing Expressions and Equations

An expression has no equals sign.

An equation contains an equals sign showing that two quantities have the same value.

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Expressions Are Evaluated

We usually evaluate an expression by finding its value for a particular value of the variable.

Consider:

3x + 2

If:

x = 4

then substitute 4 for x:

3(4) + 2

Multiply:

12 + 2

Therefore:

3x + 2 = 14 when x = 4

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Equations Are Solved

An equation can be solved to determine which value of the variable makes the equation true.

Consider:

3x + 2 = 14

We can solve this equation to find:

x = 4

So:

Expressions are evaluated

while:

Equations are solved

This distinction becomes increasingly important as you progress through algebra.

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Interpreting Expressions

Algebraic expressions can represent real situations.

Consider:

5n

Suppose n represents the number of notebooks purchased and each notebook costs $5.

Then:

5n

represents:

The total cost of n notebooks

The meaning of an expression depends on what the variable represents.

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Worked Example — Taxi Fare

Suppose a taxi charges:

$4 starting fee

and:

$3 for each kilometre travelled

Let:

d = distance travelled in kilometres

The distance charge is:

3d

The total cost is:

3d + 4

So the expression:

3d + 4

can be interpreted as:

$3 for every kilometre travelled plus a fixed $4 starting fee.

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Worked Example — Buying Tickets

A theme park charges $15 per ticket.

Let:

t = number of tickets

The total ticket cost is:

15t

If there is also a fixed $5 booking fee:

15t + 5

The coefficient 15 represents the cost per ticket.

The variable t represents the number of tickets.

The constant 5 represents the fixed booking fee.

This demonstrates how the different parts of an expression can have real-world meanings.

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Worked Example — Temperature

Suppose the temperature is currently T degrees and then rises by 6 degrees.

The new temperature can be represented by:

T + 6

If instead the temperature decreases by 6 degrees:

T − 6

Variables allow the same expression to describe many possible starting temperatures.

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Like and Unlike Terms

Terms containing exactly the same variable part are called like terms.

For example:

3x

and:

7x

are like terms.

So are:

2y

and:

−5y

But:

3x

and:

3y

are unlike terms because they contain different variables.

Similarly:

x

and:

x²

are unlike terms because the variable parts are different.

Recognizing terms correctly is important when simplifying expressions later.

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Algebraic Expressions in Geometry

Expressions can also describe measurements.

Suppose a rectangle has:

Length = x + 3

and:

Width = x

Its perimeter can be represented using:

P = 2(length) + 2(width)

Substitute the expressions:

P = 2(x + 3) + 2x

Algebra therefore allows us to represent geometric relationships even when dimensions are unknown.

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Interpreting an Expression Step by Step

Consider:

8n + 12

Suppose n represents the number of people attending an event.

We should ask:

What does 8 represent?

It could represent a cost of $8 per person.

What does n represent?

The number of people.

What does 8n represent?

The variable cost for all the people.

What does 12 represent?

A fixed additional charge.

Therefore:

8n + 12

could represent a total cost consisting of $8 per person plus a fixed $12 charge.

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Why Algebraic Expressions Are Useful

Algebraic expressions allow us to describe patterns and relationships efficiently.

Instead of writing:

Cost for 1 item = $5

Cost for 2 items = $10

Cost for 3 items = $15

Cost for 4 items = $20

we can simply write:

Cost = 5n

where n is the number of items.

One expression can describe an unlimited number of situations.

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Common Mistakes

Mistake: Confusing a coefficient with a constant

In:

5x + 8

5 is the coefficient of x.

8 is the constant.

Mistake: Forgetting hidden coefficients

In:

x + 4

the coefficient of x is:

1

Mistake: Forgetting the negative sign

In:

3x − 7

the constant term is:

−7

Mistake: Reversing "less than"

5 less than x

means:

x − 5

not:

5 − x

Mistake: Confusing an expression with an equation

4x + 3

is an expression.

4x + 3 = 15

is an equation.

Mistake: Writing 4 + x when the phrase says four times x

Four times x

means:

4x

not:

4 + x

Mistake: Ignoring grouping words

Twice the sum of x and 3

means:

2(x + 3)

not:

2x + 3

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Key Terms

Algebra – a branch of mathematics that uses symbols to represent numbers and relationships.

Expression – a mathematical phrase containing numbers, variables, and operations.

Variable – a letter or symbol representing a number.

Term – one part of an expression separated by addition or subtraction.

Coefficient – a numerical factor multiplying a variable.

Constant – a value that does not change.

Equation – a mathematical statement showing that two expressions have equal values.

Evaluate – calculate the value of an expression for particular variable values.

Algebraic notation – the mathematical symbols and conventions used to represent relationships.

Like terms – terms containing the same variable raised to the same power.

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Check Your Understanding

1. Identify the variable, coefficient, and constant in:

5x + 9

2. List the terms in:

3a + 7b − 4

3. What is the coefficient of x in:

−6x + 2

4. What is the coefficient of y in:

y + 8

5. Write an expression for:

Seven more than x

6. Write an expression for:

Five times n

7. Write an expression for:

Four less than y

8. Write an expression for:

Three times the sum of x and 2

9. Which of the following is an expression?

4x + 7

or

4x + 7 = 19

Explain your answer.

10. A cinema ticket costs $12. Write an expression for the cost of t tickets.

11. A delivery company charges $5 plus $2 per kilometre. If d represents the distance in kilometres, write an expression for the total cost.

12. Interpret the meaning of:

7n + 10

if n represents the number of students and each student pays $7.

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Key Takeaways

• An algebraic expression contains numbers, variables, and mathematical operations.
• A variable represents a value that can change or may be unknown.
• A term is one part of an expression separated by addition or subtraction.
• A coefficient is the numerical factor multiplying a variable.
• A constant is a number without a variable.
• Mathematical words and phrases can be translated into algebraic notation.
• Words such as sum, difference, product, quotient, more than, and less than provide clues about which operations to use.
• Order matters when translating subtraction phrases such as "less than."
• Brackets are sometimes necessary to represent grouped operations correctly.
• An expression does not contain an equals sign.
• An equation states that two expressions are equal.
• Expressions can represent real-world quantities such as costs, distances, temperatures, and measurements.
• Understanding what each term represents allows us to interpret the meaning of an algebraic expression.

Did You Know?

The word algebra comes from the Arabic word al-jabr, which appeared in the title of a mathematical work written by the scholar al-Khwarizmi around the ninth century. His work helped establish systematic methods for solving equations and strongly influenced the development of modern algebra.