Combining Like Terms

Algebraic expressions often contain several terms. Some of these terms represent the same type of quantity, which means they can be combined to make the expression simpler.

For example:

3x + 5x

contains two like terms. Since both terms contain x, they can be combined:

3x + 5x = 8x

Learning to identify and combine like terms is an important step toward simplifying expressions and eventually solving algebraic equations.

Learning Targets

• I can identify like terms.
• I can combine like terms correctly.
• I can simplify expressions containing several terms.
• I can recognize equivalent expressions.
• I can check whether two expressions are equivalent.

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What Is a Term?

A term is one part of an algebraic expression.

Terms are separated by addition or subtraction signs.

For example:

4x + 3y − 7

contains three terms:

4x

3y

−7

Each term may contain:

• A coefficient
• A variable
• An exponent
• Or simply a constant

In the term:

5x

5 is the coefficient and x is the variable.

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What Are Like Terms?

Like terms are terms that have exactly the same variable part.

For example:

3x and 7x

are like terms because they both contain x.

Similarly:

4y and −2y

are like terms.

And:

5a² and 9a²

are also like terms because both contain a².

The coefficients do not need to be the same.

It is the variable part that determines whether terms are like terms.

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Unlike Terms

Terms with different variable parts are called unlike terms.

For example:

3x and 3y

are unlike terms.

Even though their coefficients are both 3, the variables are different.

Similarly:

5x and 5x²

are unlike terms.

The exponent is part of the variable portion of the term.

Therefore:

x and x² are different types of terms.

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Identifying Like Terms

Consider:

4x + 7y + 2x − 3

The terms are:

4x

7y

2x

−3

The like terms are:

4x and 2x

because they both contain x.

There is only one y term, so it cannot be combined with anything else.

The constant −3 also has no other constant to combine with.

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Constants Are Like Terms

Numbers without variables are called constants.

All constants are like terms with one another.

For example:

3x + 7 + 2x + 5

The x terms are:

3x and 2x

The constants are:

7 and 5

Therefore, both pairs can be combined.

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Why Can Like Terms Be Combined?

Imagine you have:

3 apples + 5 apples

You can combine them:

3 apples + 5 apples = 8 apples

But:

3 apples + 5 oranges

cannot be written as:

8 apples

or:

8 oranges

The quantities are different.

Algebra works in the same way.

3x + 5x = 8x

but:

3x + 5y

cannot be combined.

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Algebra Tiles

Algebra tiles provide another way to visualize like terms.

Imagine that each long rectangular tile represents:

x

Three x-tiles represent:

3x

If we add another two x-tiles:

3x + 2x

we now have five x-tiles:

5x

Algebra tiles help show why only matching types of terms can be combined.

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Combining Like Terms

To combine like terms:

• Identify terms with identical variable parts.
• Look at their coefficients.
• Add or subtract the coefficients.
• Keep the variable part unchanged.

For example:

4x + 3x

Add the coefficients:

4 + 3 = 7

Keep x:

7x

Therefore:

4x + 3x = 7x

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Worked Example — Addition

Simplify:

6y + 2y

Both terms contain y.

Add the coefficients:

6 + 2 = 8

Therefore:

6y + 2y = 8y

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Worked Example — Subtraction

Simplify:

9x − 4x

Both terms contain x.

Subtract the coefficients:

9 − 4 = 5

Therefore:

9x − 4x = 5x

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Negative Coefficients

Like terms can have negative coefficients.

For example:

7x − 10x

Think of this as:

7x + (−10x)

Combine the coefficients:

7 + (−10) = −3

Therefore:

7x − 10x = −3x

The rules for positive and negative numbers still apply.

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Hidden Coefficients

Remember that a variable written without a coefficient has a coefficient of 1.

Therefore:

x = 1x

and:

−x = −1x

Consider:

x + 4x

This means:

1x + 4x

Therefore:

5x

Similarly:

6y − y

means:

6y − 1y

Therefore:

5y

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Combining Constants

Constants can also be combined.

Consider:

3x + 5 + 8

The constants are:

5 and 8

Combine them:

5 + 8 = 13

Therefore:

3x + 5 + 8 = 3x + 13

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Simplifying Expressions with Several Terms

Expressions often contain more than one group of like terms.

Consider:

3x + 5y + 4x + 2y

Identify the x terms:

3x + 4x = 7x

Identify the y terms:

5y + 2y = 7y

Therefore:

3x + 5y + 4x + 2y = 7x + 7y

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Organizing Like Terms

When an expression contains many terms, it can help to group like terms together.

For example:

5x + 3y + 2 + 4x − y + 6

Group the x terms:

5x + 4x

Group the y terms:

3y − y

Group the constants:

2 + 6

Now combine:

9x + 2y + 8

Therefore:

5x + 3y + 2 + 4x − y + 6 = 9x + 2y + 8

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Worked Example — Several Terms

Simplify:

8a + 4b − 3a + 7b

Combine the a terms:

8a − 3a = 5a

Combine the b terms:

4b + 7b = 11b

Therefore:

8a + 4b − 3a + 7b = 5a + 11b

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Worked Example — Including Constants

Simplify:

6x + 4 − 2x + 9

Combine the x terms:

6x − 2x = 4x

Combine the constants:

4 + 9 = 13

Therefore:

6x + 4 − 2x + 9 = 4x + 13

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Worked Example — Negative Terms

Simplify:

10m − 3n − 6m + 8n

Combine the m terms:

10m − 6m = 4m

Combine the n terms:

−3n + 8n = 5n

Therefore:

10m − 3n − 6m + 8n = 4m + 5n

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Different Variables Cannot Be Combined

Consider:

4x + 3y

These terms cannot be combined.

There is no simpler expression.

Therefore:

4x + 3y

is already simplified.

This is similar to saying:

4 metres + 3 kilograms

The quantities represent different things and cannot simply be added together.

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Exponents Must Match

Variables must have the same exponents for terms to be like terms.

Consider:

3x + 5x²

These are not like terms.

One contains:

x

The other contains:

x²

Therefore:

3x + 5x²

cannot be simplified by combining the terms.

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Terms with Several Variables

The same rule applies when terms contain multiple variables.

For example:

3xy and 7xy

are like terms.

Therefore:

3xy + 7xy = 10xy

However:

3xy and 3x

are not like terms.

Similarly:

4x²y and 9x²y

are like terms because their variable parts are identical.

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Worked Example — Exponents

Simplify:

4x² + 3x + 6x² − x

Group the x² terms:

4x² + 6x² = 10x²

Group the x terms:

3x − x = 2x

Therefore:

4x² + 3x + 6x² − x = 10x² + 2x

The x² and x terms remain separate.

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What Does Simplify Mean?

To simplify an expression means to rewrite it in a shorter or clearer equivalent form.

For example:

3x + 2x + 7

can be simplified to:

5x + 7

The expressions look different, but they represent the same value for every possible value of x.

That means they are equivalent expressions.

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Equivalent Expressions

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

For example:

3x + 4x

and:

7x

are equivalent.

We can write:

3x + 4x = 7x

Similarly:

5y + 3 − 2y + 4

simplifies to:

3y + 7

Therefore:

5y + 3 − 2y + 4

and:

3y + 7

are equivalent expressions.

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Checking Equivalent Expressions by Simplifying

One way to determine whether two expressions are equivalent is to simplify them.

Consider:

Expression A: 4x + 3x + 5

Expression B: 7x + 5

Simplify Expression A:

4x + 3x + 5

= 7x + 5

This is identical to Expression B.

Therefore, the expressions are equivalent.

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Checking by Substitution

Another useful method is to substitute values for the variable.

Consider:

Expression A: 2x + 3x

Expression B: 5x

Choose:

x = 4

Expression A:

2(4) + 3(4)

= 8 + 12

= 20

Expression B:

5(4)

= 20

Both give the same result.

This provides evidence that they are equivalent.

However, testing only one value is not enough to prove that two expressions are always equivalent. Simplifying algebraically is a stronger method.

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Why One Test Value Is Not Enough

Consider:

Expression A: x + 2

Expression B: 2x

If:

x = 2

Expression A:

2 + 2 = 4

Expression B:

2(2) = 4

They match.

But try:

x = 5

Expression A:

5 + 2 = 7

Expression B:

2(5) = 10

They do not match.

Therefore, the expressions are not equivalent.

A single matching value can happen by coincidence.

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Worked Example — Checking Equivalence

Are these expressions equivalent?

A = 7x + 2 − 3x + 5

B = 4x + 7

Simplify A:

7x − 3x = 4x

and:

2 + 5 = 7

Therefore:

A = 4x + 7

This matches B exactly.

Therefore:

The expressions are equivalent.

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Combining Like Terms with Decimals

The coefficients do not need to be whole numbers.

For example:

2.5x + 1.5x

Combine the coefficients:

2.5 + 1.5 = 4

Therefore:

2.5x + 1.5x = 4x

Another example:

6.2y − 1.7y

= 4.5y

The same rules apply.

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Combining Like Terms with Fractions

Fractions can also be coefficients.

For example:

1/2 x + 1/4 x

The variable parts match, so combine the coefficients:

1/2 + 1/4 = 3/4

Therefore:

1/2 x + 1/4 x = 3/4 x

The algebraic rule is unchanged; only the arithmetic with the coefficients is different.

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Combining Terms in Real-World Situations

Suppose you buy:

3 notebooks at x dollars each

and later buy:

5 more notebooks at x dollars each

The first purchase costs:

3x

The second purchase costs:

5x

Total cost:

3x + 5x

Combine the like terms:

8x

The simplified expression tells us that buying the two groups is equivalent to buying eight notebooks at x dollars each.

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Combining Distances

Suppose one section of a journey has length:

4d

and another has length:

3d

The total distance is:

4d + 3d

Therefore:

7d

But if another part of the problem represents time as t, then:

7d + 2t

cannot be combined further because d and t represent different quantities.

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Checking Your Work

After simplifying an expression, ask:

Did I identify every term correctly?

Did I include negative signs with their terms?

Do the variables match exactly?

Do the exponents match?

Did I combine only the coefficients?

Did I keep the variable part unchanged?

Did I combine the constants?

A quick check can prevent many algebra mistakes.

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Common Mistakes

Mistake: Combining unlike terms

3x + 4y

does not equal:

7xy

and it does not equal:

7x

The terms are unlike and must remain separate.

Mistake: Adding the variables

3x + 4x

does not equal:

7x²

The correct answer is:

7x

Only the coefficients are added.

Mistake: Ignoring exponents

2x + 5x²

cannot be combined.

The variable parts are different.

Mistake: Losing a negative sign

In:

8x − 5x

the second coefficient is:

−5

Therefore:

8 + (−5) = 3

and the result is:

3x

Mistake: Forgetting hidden coefficients

x + 3x

means:

1x + 3x

Therefore:

4x

Mistake: Forgetting to combine constants

3x + 4 + 2x + 7

should simplify completely to:

5x + 11

not just:

5x + 4 + 7

Mistake: Assuming one matching substitution proves equivalence

Two different expressions can sometimes give the same answer for one particular value.

To establish equivalence, simplify the expressions algebraically or use a valid algebraic argument.

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Key Terms

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

Like terms – terms with identical variable parts.

Unlike terms – terms with different variable parts.

Coefficient – the numerical factor multiplying a variable.

Variable – a letter or symbol representing a number.

Constant – a number without a variable.

Simplify – rewrite an expression in a simpler equivalent form.

Equivalent expressions – expressions that have the same value for every allowed value of their variables.

Substitution – replacing a variable with a particular value.

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Check Your Understanding

1. Identify the like terms in:

3x + 5y + 7x

2. Simplify:

4x + 6x

3. Simplify:

9y − 3y

4. Simplify:

5a + 3b + 2a

5. Simplify:

7x + 4 + 2x + 6

6. Simplify:

8m − 3n − 5m + 7n

7. Simplify:

4x² + 2x + 3x² + 5x

8. Explain why:

3x + 5y

cannot be simplified to 8xy.

9. Are:

4x + 3x + 2

and:

7x + 2

equivalent? Explain.

10. Are:

3x + 6

and:

6x + 3

equivalent? Explain.

11. Simplify:

x + 5x − 2x

12. Simplify:

2.5y + 4y − 1.5y

13. Simplify:

5a + 3 + 2b − 2a + 8 − b

14. Explain why checking only one value of x is not enough to prove that two expressions are equivalent.

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Key Takeaways

• Like terms have exactly the same variable parts.
• Coefficients can be different while terms are still like terms.
• Variables and their exponents must match.
• Constants are like terms with other constants.
• To combine like terms, add or subtract the coefficients and keep the variable part unchanged.
• Unlike terms cannot be combined.
• Hidden coefficients such as 1x and −1x must be considered.
• Expressions containing several variables can be simplified by grouping each set of like terms.
• x and x² are not like terms.
• Simplifying an expression does not change its value.
• Equivalent expressions have the same value for every allowed value of their variables.
• Expressions can be checked for equivalence by simplifying them.
• Substitution is useful for checking examples, but one matching value alone does not prove equivalence.
• Combining like terms makes algebraic expressions shorter, clearer, and easier to work with.

Did You Know?

Computer algebra systems use the same basic idea when simplifying enormous mathematical expressions. A computer may process thousands or millions of terms, identify terms with matching algebraic structures, and combine them automatically. The principle is the same as simplifying 3x + 5x into 8x—just on a much larger scale.