Combining Like Terms

Combining like terms is one of the most important skills used to simplify algebraic expressions.

Consider:

3x + 5x

Both terms contain the same variable, x, so they are like terms.

We can combine them:

3x + 5x = 8x

This works because we are really saying:

3 groups of x + 5 groups of x = 8 groups of x

Just as:

3 apples + 5 apples = 8 apples

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What Is a Term?

A term is a number, variable, or product of numbers and variables within an expression.

Terms are separated by addition or subtraction.

For example:

4x + 7 − 2y

contains three terms:

4x

7

−2y

Notice that the negative sign belongs to the term:

−2y

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Parts of a Term

Consider:

6x

The number 6 is the:

coefficient

The letter x is the:

variable

In:

−4y

the coefficient is:

−4

and the variable is:

y

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Constants

A constant is a term containing no variable.

Examples:

5

−3

12

1.7

Constants are like terms with other constants.

For example:

7 + 4 = 11

and:

12 − 5 = 7

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What Are Like Terms?

Like terms contain exactly the same variable part.

Examples:

3x and 8x

are like terms.

5y and −2y

are like terms.

4a² and 9a²

are like terms.

3xy and −7xy

are also like terms.

Only their coefficients are different.

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Identifying Like Terms

Consider:

4x + 3 + 7x − 5

The like terms are:

4x and 7x

and:

3 and −5

Therefore:

4x + 3 + 7x − 5

can be reorganized as:

4x + 7x + 3 − 5

Then:

11x − 2

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Unlike Terms

Terms are unlike terms if their variable parts are different.

Examples:

3x and 3y

are unlike.

4x and 4x²

are unlike.

2xy and 2x

are unlike.

5a² and 5a

are unlike.

These terms cannot be combined by simply adding or subtracting their coefficients.

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Why Can Only Like Terms Be Combined?

Consider:

3 apples + 4 apples

These can be combined:

7 apples

But:

3 apples + 4 oranges

cannot become:

7 apples

or:

7 oranges

They are different types of objects.

Algebra works in the same way.

3x + 4x = 7x

but:

3x + 4y

cannot be simplified further because x and y represent different quantities.

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Combining Coefficients

When combining like terms, we combine the coefficients but keep the variable part unchanged.

For example:

4x + 7x

Combine:

4 + 7 = 11

Keep x:

11x

Therefore:

4x + 7x = 11x

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Why the Variable Does Not Change

Remember:

4x = x + x + x + x

and:

3x = x + x + x

So:

4x + 3x

means:

x + x + x + x + x + x + x

which is:

7x

The x does not become x².

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Subtracting Like Terms

Like terms can also be subtracted.

Example:

9x − 4x

Subtract the coefficients:

9 − 4 = 5

Therefore:

9x − 4x = 5x

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Negative Coefficients

Consider:

3x − 8x

Think of this as:

3x + (−8x)

Combine the coefficients:

3 + (−8) = −5

Therefore:

3x − 8x = −5x

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Combining Two Negative Terms

Consider:

−4y − 6y

Combine:

−4 + (−6) = −10

Therefore:

−4y − 6y = −10y

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When Terms Cancel

Consider:

7x − 7x

Combine coefficients:

7 − 7 = 0

So:

0x = 0

Therefore:

7x − 7x = 0

The variable terms completely cancel.

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The Invisible Coefficient

A variable written without a number has a coefficient of:

1

So:

x = 1x

and:

−x = −1x

This is useful when combining terms.

Example:

x + 5x

becomes:

1x + 5x = 6x

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Worked Example

Simplify:

5x + 3x

The terms are like because both contain x.

Add coefficients:

5 + 3 = 8

Therefore:

5x + 3x = 8x

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Worked Example with Subtraction

Simplify:

12a − 7a

Combine coefficients:

12 − 7 = 5

Therefore:

12a − 7a = 5a

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Worked Example with Constants

Simplify:

4x + 7 + 3x + 2

Group like terms:

4x + 3x + 7 + 2

Combine:

7x + 9

Therefore:

4x + 7 + 3x + 2 = 7x + 9

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Rearranging Terms

The order of terms can be changed when addition is involved.

For example:

3x + 7 + 5x + 2

can be rearranged as:

3x + 5x + 7 + 2

Then:

8x + 9

This makes the like terms easier to see.

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Keep the Sign with the Term

Consider:

8x − 3 + 2x − 7

The terms are:

8x

−3

2x

−7

Group them:

8x + 2x − 3 − 7

Combine:

10x − 10

Keeping the sign attached to each term helps prevent errors.

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Expressions with Two Variables

Simplify:

4x + 3y + 2x + 5y

Group x terms:

4x + 2x

Group y terms:

3y + 5y

Combine:

6x + 8y

The x and y terms cannot be combined with each other.

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Worked Example with Two Variables

Simplify:

7a + 2b − 3a + 5b

Group:

7a − 3a + 2b + 5b

Combine:

4a + 7b

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Three Types of Terms

Consider:

3x + 4y + 5 + 2x − y − 8

There are three groups of like terms.

x terms:

3x + 2x = 5x

y terms:

4y − y = 3y

constants:

5 − 8 = −3

Therefore:

5x + 3y − 3

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Powers Matter

Terms are only like terms when the variables and their exponents match.

For example:

3x² + 5x² = 8x²

But:

3x² + 5x

cannot be combined.

Even though both contain x, their variable parts are different:

x² ≠ x

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Example with Powers

Simplify:

4x² + 3x + 5x² − x

Group:

4x² + 5x² + 3x − x

Combine:

9x² + 2x

The x² terms combine with each other.

The x terms combine with each other.

But x² and x do not combine.

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Multiple Variables

The complete variable part must match.

For example:

3xy + 5xy = 8xy

But:

3xy + 5x

cannot be combined.

Similarly:

4x²y + 7x²y = 11x²y

but:

4x²y + 7xy²

cannot be combined because the exponents are different.

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A Systematic Method

When simplifying an expression:

Identify → Group → Combine → Rewrite

Identify

Find terms with identical variable parts.

Group

Place like terms together.

Combine

Add or subtract their coefficients.

Rewrite

Write the simplified expression clearly.

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Worked Example Using the Method

Simplify:

6x + 4 − 2x + 9

Identify

Like variable terms:

6x and −2x

Like constants:

4 and 9

Group

6x − 2x + 4 + 9

Combine

4x + 13

Therefore:

6x + 4 − 2x + 9 = 4x + 13

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Expressions with Parentheses

Sometimes we need to use the distributive property before combining like terms.

Consider:

3(x + 4) + 2x

Distribute:

3x + 12 + 2x

Now combine:

3x + 2x = 5x

Therefore:

5x + 12

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Distributive Property

The distributive property states:

a(b + c) = ab + ac

Example:

4(x + 3)

becomes:

4x + 12

Once the parentheses are removed, we can identify and combine like terms.

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Worked Example with Distribution

Simplify:

2(x + 5) + 3x + 4

Distribute:

2x + 10 + 3x + 4

Group:

2x + 3x + 10 + 4

Combine:

5x + 14

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Subtracting Parentheses

Be careful when a negative number is outside parentheses.

Simplify:

5x − 2(x + 3)

Distribute −2:

5x − 2x − 6

Combine:

3x − 6

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Negative Distribution

Simplify:

−3(x − 4) + 2x

Distribute −3:

−3x + 12 + 2x

Combine:

−x + 12

Remember:

−3 × −4 = +12

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Equivalent Expressions

Two expressions are equivalent if they have the same value for every permissible value of the variables.

For example:

3x + 5x

and:

8x

are equivalent.

We write:

3x + 5x = 8x

because both expressions always produce the same result.

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Checking Equivalent Expressions

Consider:

4x + 2x + 3

and:

6x + 3

Test x = 5.

First expression:

4(5) + 2(5) + 3

= 20 + 10 + 3

= 33

Second expression:

6(5) + 3

= 30 + 3

= 33

They match.

Testing values can provide evidence that expressions are equivalent, while correct algebra shows why they are equivalent for all permissible values.

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Equivalent Does Not Mean Identical Looking

Consider:

2(x + 3)

and:

2x + 6

They look different, but distributing gives:

2(x + 3) = 2x + 6

Therefore they are equivalent expressions.

Similarly:

5x + 3x − 2

and:

8x − 2

are equivalent.

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Expressions Versus Equations

An expression does not contain an equals sign as part of the expression itself.

Example:

3x + 5x + 2

We simplify it to:

8x + 2

An equation states that two expressions are equal.

Example:

3x + 5x + 2 = 18

Simplify:

8x + 2 = 18

Combining like terms is often one of the first steps in solving equations.

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Combining Like Terms in Equations

Solve:

3x + 5x + 4 = 28

First combine:

8x + 4 = 28

Subtract 4:

8x = 24

Divide by 8:

x = 3

Combining like terms makes the equation easier to solve.

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Like Terms on Each Side

Consider:

3x + 2x + 4 = x + 16

First simplify the left:

5x + 4 = x + 16

Subtract x:

4x + 4 = 16

Subtract 4:

4x = 12

Therefore:

x = 3

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Combining Like Terms in Geometry

Suppose the sides of a triangle are:

2x + 3

4x

x + 5

The perimeter can be written:

2x + 3 + 4x + x + 5

Combine x terms:

2x + 4x + x = 7x

Combine constants:

3 + 5 = 8

Therefore the perimeter is:

7x + 8

This expression is simpler and easier to use.

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Combining Like Terms in Real Life

Suppose a student buys:

• 3 notebooks costing x dollars each
• 4 more notebooks costing x dollars each
• a pen costing $2

The total cost is:

3x + 4x + 2

Combine like terms:

7x + 2

This simplified expression represents the same total cost.

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Algebra Tiles

Algebra tiles can help visualize why like terms combine.

Different tiles represent different types of terms.

For example:

• long rectangular tiles can represent x
• large square tiles can represent x²
• small squares can represent 1

Three x tiles and two x tiles can combine to make:

5x

But an x tile and an x² tile represent different quantities, so they cannot be combined into a single term.

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Why 3x + 4 Is Not 7x

This is one of the most common algebra mistakes.

Consider:

3x + 4

The first term means:

3 × x

The second term is simply:

4

They are different types of terms.

If x = 2:

Original expression:

3(2) + 4 = 10

If we incorrectly wrote 7x:

7(2) = 14

Since:

10 ≠ 14

the expressions are not equivalent.

Therefore:

3x + 4 cannot be simplified to 7x.

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Why 3x + 4x Is Not 7x²

Another common mistake is changing the exponent.

Correct:

3x + 4x = 7x

Not:

7x²

Remember that adding terms does not multiply their variables.

For x = 2:

3(2) + 4(2) = 14

and:

7(2) = 14

But:

7(2²) = 28

So 7x² is not equivalent.

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Addition and Multiplication Behave Differently

Compare:

x + x = 2x

but:

x × x = x²

Similarly:

3x + 4x = 7x

while:

(3x)(4x) = 12x²

Combining like terms applies to addition and subtraction, not multiplication in the same way.

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Simplifying a Longer Expression

Simplify:

5x² + 3x − 7 + 2x² − 8x + 10

Group x² terms:

5x² + 2x² = 7x²

Group x terms:

3x − 8x = −5x

Group constants:

−7 + 10 = 3

Therefore:

7x² − 5x + 3

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Worked Example with Several Variables

Simplify:

4a + 3b − 2a + 7 − 5b + 4

Group a terms:

4a − 2a = 2a

Group b terms:

3b − 5b = −2b

Group constants:

7 + 4 = 11

Therefore:

2a − 2b + 11

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Worked Example with Powers

Simplify:

6x² − 2x + 4 + 3x² + 5x − 9

x² terms:

6x² + 3x² = 9x²

x terms:

−2x + 5x = 3x

constants:

4 − 9 = −5

Therefore:

9x² + 3x − 5

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Worked Example with Distribution

Simplify:

3(2x + 5) − 4x + 7

Distribute:

6x + 15 − 4x + 7

Combine x terms:

6x − 4x = 2x

Combine constants:

15 + 7 = 22

Therefore:

2x + 22

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Worked Example with Two Sets of Parentheses

Simplify:

2(x + 3) + 4(x − 1)

Distribute:

2x + 6 + 4x − 4

Combine:

6x + 2

Therefore:

2(x + 3) + 4(x − 1) = 6x + 2

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Simplifying Does Not Mean Solving

If asked to simplify:

3x + 5x − 4

the answer is:

8x − 4

We cannot find x because there is no equation giving enough information to determine x.

Simplifying means rewriting an expression in a simpler equivalent form.

Solving means finding the value or values of a variable that make an equation true.

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A Useful Organization Strategy

For longer expressions, identify categories.

For:

3x² + 4x − 7 + 2x² − x + 9

create groups:

x² terms: 3x² + 2x²

x terms: 4x − x

constants: −7 + 9

Then combine:

5x² + 3x + 2

This reduces sign and classification mistakes.

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Common Mistakes

Mistake: Combining unlike variables.

Incorrect:

3x + 4y = 7xy

Correct:

3x + 4y

cannot be simplified further.

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Mistake: Combining variables and constants.

Incorrect:

5x + 3 = 8x

Correct:

5x + 3

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Mistake: Combining different powers.

Incorrect:

2x² + 3x = 5x³

Correct:

2x² + 3x

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Mistake: Changing the exponent when adding.

Incorrect:

4x + 2x = 6x²

Correct:

4x + 2x = 6x

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Mistake: Losing negative signs.

In:

7x − 3x − 5

the terms are:

7x, −3x, −5

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Mistake: Forgetting an invisible coefficient.

x + 4x

means:

1x + 4x = 5x

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Mistake: Distributing to only one term.

Incorrect:

3(x + 4) = 3x + 4

Correct:

3(x + 4) = 3x + 12

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Mistake: Confusing addition and multiplication.

Remember:

x + x = 2x

but:

x × x = x²

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Key Terms

Term — number, variable, or product of numbers and variables within an expression.

Like terms — terms with exactly the same variable part, including exponents.

Unlike terms — terms with different variable parts.

Coefficient — numerical factor multiplying a variable.

Variable — letter or symbol representing a number or quantity.

Constant — term containing no variable.

Expression — combination of numbers, variables, and mathematical operations.

Simplify — rewrite an expression in a simpler equivalent form.

Combine like terms — add or subtract the coefficients of terms with identical variable parts.

Equivalent expressions — expressions having the same value for every permissible value of their variables.

Exponent — number indicating how many times a quantity is used as a factor.

Distributive property — property represented by a(b + c) = ab + ac.

Algebra tiles — visual models representing constants, variables, and powers of variables.

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Key Takeaways

• Expressions are made of terms.
• Addition and subtraction signs separate terms.
• The sign immediately before a term belongs to that term.
• Like terms have exactly the same variable part.
• The variables and their exponents must match.
• Constants are like terms with other constants.
• Coefficients may be different while terms are still like.
• Only like terms can be combined through addition or subtraction.
• Combining like terms means combining their coefficients.
• The variable part remains unchanged.
• 3x + 5x = 8x.
• 3x + 5 cannot be combined.
• 3x + 5y cannot be combined.
• 3x² + 5x² = 8x².
• 3x² + 5x cannot be combined.
• x has an implied coefficient of 1.
• −x has an implied coefficient of −1.
• Opposite like terms can cancel completely.
• The distributive property may need to be used before combining like terms.
• Simplifying does not change the value represented by an expression.
• Equivalent expressions can look different while representing the same quantity.
• Combining like terms is essential for simplifying expressions and solving equations.

Check Your Understanding

1. What is a term?
2. What makes two terms "like terms"?
3. Identify the coefficient in 7x.
4. Identify the terms in 4x − 3y + 8.
5. Which terms are like in 3x + 5y + 7x − 2y?
6. Simplify: 4x + 7x.
7. Simplify: 12y − 5y.
8. Simplify: 3a + 7 + 5a + 4.
9. Simplify: 8x − 3x + 5 − 9.
10. Simplify: 4x + 3y + 2x − 5y.
11. Simplify: 5x² + 3x² − 2x.
12. Explain why 4x² + 3x cannot be combined into one term.
13. Simplify: 3(x + 4) + 2x.
14. Simplify: 5(x − 2) − 3x + 7.
15. Simplify: −2(x − 4) + 5x.
16. Simplify: 4x² + 3x − 7 + 2x² − 5x + 10.
17. Are 3x + 5x and 8x equivalent? Explain why.
18. Explain why 3x + 4 cannot be simplified to 7x.
19. Explain the difference between x + x and x × x.
20. Describe a systematic method you can use to identify and combine like terms in a long algebraic expression.