Introduction
Look at this simple expression: 3x + 5x
Both terms contain exactly the same variable, x, raised to the same power. Because of that, they can be added together. The result is 8x.
So, how to combine like terms? You add or subtract the numerical coefficients of the terms while keeping the variable part completely unchanged. That is the entire process. The variable is not added, multiplied, or changed in any way. Only the numbers in front of it change.
Combining like terms is one of the most fundamental skills in algebra. It is used to simplify algebraic expressions, clean up both sides of equations, and prepare expressions for factoring, solving, or graphing. Once you understand the method clearly, it becomes fast and reliable.
Key Takeaways
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Like terms have identical variable parts, meaning the same variables raised to the same exponents.
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To combine like terms, add or subtract their coefficients while keeping the variable part the same.
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Unlike terms cannot be combined because their variable parts are different.
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Negative signs belong to the coefficient of each term and must not be detached.
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Constants are like terms with other constants and can always be combined.
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Exponents are never added when combining like terms. Only coefficients change.
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Combining like terms is a simplification step, not the same as solving an equation.
What Does It Mean to Combine Like Terms?
Combining like terms means simplifying an algebraic expression by adding or subtracting terms that share the same variable part.
When you combine like terms, you are performing arithmetic on the coefficients. The variable part is simply carried forward unchanged.
Example: 3x + 5x
The variable part in both terms is x. The coefficients are 3 and 5.
Add the coefficients: 3 + 5 = 8
Keep the variable: x
Result: 8x
Think of it this way. If you have 3 books and then receive 5 more books, you have 8 books. The unit (books) does not change. Only the quantity does. In algebra, x is the unit, and the coefficient is the quantity.
What Are Like Terms?
Like terms are algebraic terms that have exactly the same variables, each raised to exactly the same exponent. Their coefficients can be any numbers at all.
Examples of like terms:
- 3x and 7x — same variable x, same exponent 1
- 4x² and 9x² — same variable x, same exponent 2
- 2ab and 5ab — same variables a and b, same exponents 1 and 1
The coefficients (3, 7, 4, 9, 2, 5) differ across these pairs, but that is perfectly fine. It is the variable part that determines whether terms are like.
For a full explanation of what makes terms like or unlike, along with many more examples, the article [What Are Like Terms in Algebra?] covers everything you need to know.
What Are Unlike Terms?
Unlike terms are terms that do not share the same variable part. They have different variables, different exponents, or a different combination of variables.
Examples of unlike terms:
- 3x and 5y — different variables (x and y)
- 4x and 2x² — same variable but different exponents (1 and 2)
- 3ab and 5ac — different variable combinations (ab versus ac)
Unlike terms cannot be simplified into a single term. In an expression such as 3x + 5y, you cannot merge these two terms. They must stay separate.
The article [What Are Unlike Terms in Algebra?] provides a complete explanation of what makes terms unlike, including examples with exponents, multiple variables, and negative coefficients.
The Basic Rule for Combining Like Terms
The rule is straightforward:
New coefficient = first coefficient ± second coefficient
The variable part stays exactly the same.
Working examples:
2x + 6x:
Coefficients: 2 + 6 = 8
Variable: x
Result: 8x
9a – 4a:
Coefficients: 9 – 4 = 5
Variable: a
Result: 5a
7x² + 3x²:
Coefficients: 7 + 3 = 10
Variable: x²
Result: 10x²
The variable is never added, multiplied by the coefficient, or raised to a new power. The coefficient changes, and the variable stays as it is.
How to Combine Like Terms Step by Step
Follow this process every time:
- Identify all the terms in the expression.
- Find the like terms by comparing the variable parts.
- Group like terms together in the expression.
- Add or subtract their coefficients.
- Keep the common variable part exactly as it is.
- Write the simplified expression clearly.
- Check the answer to make sure no like terms remain uncombined.
The following sections walk through each step in detail.
Step 1: Identify the Terms
A term is a single unit in an algebraic expression, separated from other units by plus or minus signs.
Expression: 4x + 7y – 2x + 5
The terms are:
- 4x
- 7y
- -2x (the minus sign belongs to this term)
- 5 (a constant)
Always keep the sign attached to the term immediately following it. In this expression, -2x is one term with a coefficient of -2, not a positive 2.
Step 2: Find the Like Terms
Expression: 4x + 7y – 2x + 5
Compare the variable parts:
- 4x and -2x → both contain x to the power of 1. These are like terms.
- 7y → no other y term present. Stands alone.
- 5 → constant. No other constant present. Stands alone.
Step 3: Group Like Terms
Rearrange the expression so that like terms are next to each other. This reduces the chance of making errors.
From: 4x + 7y – 2x + 5
Rearranged: 4x – 2x + 7y + 5
Be careful when rearranging. Every term, including its sign, must move together.
Step 4: Combine the Coefficients
4x – 2x:
Coefficients: 4 – 2 = 2
Variable: x
Result: 2x
Now write the full simplified expression:
2x + 7y + 5
Step 5: Check the Final Expression
Look at 2x + 7y + 5.
- 2x and 7y are unlike (different variables). They stay separate.
- 2x and 5 are unlike (variable term and constant). They stay separate.
- 7y and 5 are unlike (variable term and constant). They stay separate.
The expression 2x + 7y + 5 is fully simplified. No further combining is possible.
Combining Two Like Terms
| Expression | Calculation | Result |
|---|---|---|
| 2x + 3x | 2 + 3 = 5 | 5x |
| 7a + 4a | 7 + 4 = 11 | 11a |
| 9m – 2m | 9 – 2 = 7 | 7m |
| 6y² + 5y² | 6 + 5 = 11 | 11y² |
| 10p – 7p | 10 – 7 = 3 | 3p |
| 8k + k | 8 + 1 = 9 | 9k |
| 5ab – 3ab | 5 – 3 = 2 | 2ab |
| 12n – 12n | 12 – 12 = 0 | 0 |
| 3x³ + 4x³ | 3 + 4 = 7 | 7x³ |
| 15q – 6q | 15 – 6 = 9 | 9q |
Note that when two like terms cancel each other (as in 12n – 12n), the result is simply 0, and the term disappears from the expression.
Combining Three or More Like Terms
When more than two like terms appear, combine them in order from left to right, treating each coefficient operation one at a time.
Example 1: 2x + 5x + 3x
Combine first two: 2x + 5x = 7x
Combine with third: 7x + 3x = 10x
Or simply add all coefficients: 2 + 5 + 3 = 10 → 10x
Example 2: 7a – 2a + 4a – a
Coefficients: 7 – 2 + 4 – 1 = 8 → 8a
Example 3: 4x² + x² + 3x² – 2x²
Coefficients: 4 + 1 + 3 – 2 = 6 → 6x²
Example 4: 5mn – 3mn + mn + 2mn
Coefficients: 5 – 3 + 1 + 2 = 5 → 5mn
Combining Like Terms With Negative Numbers
Negative coefficients are handled using the same rule. Keep the sign attached to its coefficient and apply standard integer arithmetic.
Example 1: -3x + 7x
Coefficients: -3 + 7 = 4
Result: 4x
Example 2: 5a – 9a
Coefficients: 5 – 9 = -4
Result: -4a
Example 3: -6y – 2y
Coefficients: -6 + (-2) = -8
Result: -8y
Example 4: -4m + (-3m)
Coefficients: -4 + (-3) = -7
Result: -7m
Example 5: 8p – 15p
Coefficients: 8 – 15 = -7
Result: -7p
Example 6: -2x² + 9x²
Coefficients: -2 + 9 = 7
Result: 7x²
Example 7: -5ab – 4ab
Coefficients: -5 – 4 = -9
Result: -9ab
Example 8: 3k – 7k + 2k
Coefficients: 3 – 7 + 2 = -2
Result: -2k
Negative results are completely valid. The coefficient becomes negative while the variable part remains unchanged.
Combining Like Terms With Positive and Negative Signs
Expressions often mix positive and negative terms. The sign in front of each term belongs to that term’s coefficient.
Example 1: 8x – 3x + 2x
Coefficients: 8 – 3 + 2 = 7
Result: 7x
Example 2: -5a + 9a – 2a
Coefficients: -5 + 9 – 2 = 2
Result: 2a
Example 3: 6y – y + 4y – 3y
Coefficients: 6 – 1 + 4 – 3 = 6
Result: 6y
Example 4: 10m – 4m – m + 2m
Coefficients: 10 – 4 – 1 + 2 = 7
Result: 7m
When you see a term written as just +x or -x with no visible number, the coefficient is +1 or -1 respectively. Never ignore these invisible 1s, as they affect the final coefficient.
Combining Like Terms With Exponents
Exponents must match exactly before you can combine terms. Only the coefficients are added or subtracted. Exponents never change during this process.
These can be combined:
3x² + 5x² = 8x² (exponents are both 2)
7a³ – 2a³ = 5a³ (exponents are both 3)
These cannot be combined:
3x² + 5x³ — exponents are 2 and 3. These are unlike terms. The expression stays as 3x² + 5x³.
A very common error is to add the exponents. This is incorrect. When combining 3x² and 5x², the result is 8x², not 8x⁴. The exponent stays exactly where it is.
Combining Like Terms With Multiple Variables
When terms contain more than one variable, the entire variable part must be identical for the terms to be like.
These can be combined:
3xy + 5xy:
Coefficients: 3 + 5 = 8. Variable part: xy.
Result: 8xy
4x²y + 2x²y:
Coefficients: 4 + 2 = 6. Variable part: x²y.
Result: 6x²y
These cannot be combined:
3xy and 5x²y:
First has xy (x to the power 1). Second has x²y (x to the power 2). The exponent on x differs. Unlike terms.
7abc and 4ab:
First contains three variables (a, b, c). Second contains two (a, b). Unlike terms.
Combining Like Terms With Fractions
When coefficients are fractions, you add or subtract the fractions using the standard method: find a common denominator and then add the numerators.
Example 1: 1/2x + 1/4x
Common denominator: 4
2/4x + 1/4x = 3/4x
Example 2: 3/5a + 2/5a
Same denominator: 5
3/5 + 2/5 = 5/5 = 1
Result: a
Example 3: 3/4m – 1/4m
Same denominator: 4
3/4 – 1/4 = 2/4 = 1/2
Result: 1/2m
Example 4: 2/3x + 1/6x
Common denominator: 6
4/6x + 1/6x = 5/6x
Result: 5/6x
For a thorough explanation of how to add and subtract fractions with different denominators, the article [What Is a Fraction?] provides all the foundational understanding you need.
Combining Like Terms With Decimal Coefficients
Decimal coefficients are handled in exactly the same way as whole-number coefficients. Add or subtract the decimal numbers.
Example 1: 0.5x + 1.5x
0.5 + 1.5 = 2.0
Result: 2x
Example 2: 2.75a – 1.25a
2.75 – 1.25 = 1.5
Result: 1.5a
Example 3: 0.3y + 0.7y
0.3 + 0.7 = 1.0
Result: y
Example 4: 4.5x – 2.3x + 0.8x
4.5 – 2.3 + 0.8 = 3.0
Result: 3x
Example 5: 1.2m – 0.4m – 0.3m
1.2 – 0.4 – 0.3 = 0.5
Result: 0.5m
Combining Like Terms With Constants
A constant is a number with no variable. All constants are like terms with each other because they all share the same empty variable part.
Examples:
- 5 + 7 = 12
- -3 + 9 = 6
- 8 – 12 = -4
- 4 + (-6) + 2 = 0
Constants cannot be combined with variable terms because they are unlike. In the expression 3x + 8, the constant 8 and the term 3x are unlike. They must stay separate.
3x + 8 stays as 3x + 8 (already simplified).
Combining Like Terms in Algebraic Expressions
Algebraic expressions often contain several different types of terms mixed together. The process is always the same: group the like terms, combine each group, and leave the unlike terms separate.
The article [What Is an Algebraic Expression?] explains the structure of algebraic expressions in full detail, which is useful background for understanding how terms fit into larger expressions.
Example: 3x + 5y + 2x – y + 7
Step 1: Identify terms: 3x, 5y, 2x, -y, 7
Step 2: Group by variable:
- x terms: 3x + 2x = 5x
- y terms: 5y – y = 4y
- Constant: 7 stays
Simplified result: 5x + 4y + 7
The three terms in the result (5x, 4y, and 7) are all unlike each other. The expression is fully simplified.
Combining Like Terms in Linear Equations
Combining like terms is often the first step when solving a linear equation. It reduces the number of terms on one or both sides of the equation, making it easier to isolate the variable.
The article [What Is a Linear Equation?] provides a complete explanation of how linear equations are structured and solved, including the role of simplification.
Example: 3x + 2x + 4 = 19
Step 1: Combine like terms on the left side.
3x + 2x = 5x
Step 2: Rewrite the equation.
5x + 4 = 19
Step 3: Solve.
5x = 15
x = 3
The simplification step comes first, and then the solving follows.
Combining Like Terms in Quadratic Expressions
Quadratic expressions can contain like terms that need to be combined before the expression is used for factoring or before an equation is solved.
The article [What Is a Quadratic Equation?] covers quadratic expressions and equations in full, including how to recognise and handle different types of terms within a quadratic.
Example: 3x² + 5x – 2x² + 7x
Step 1: Identify like terms.
- x² terms: 3x² and -2x²
- x terms: 5x and 7x
Step 2: Combine each group.
3x² – 2x² = x²
5x + 7x = 12x
Step 3: Write the final expression.
x² + 12x
Note that x² and x cannot be combined. They have the same variable but different exponents (2 and 1), making them unlike terms.
Combining Like Terms and Factors
Factors and terms are frequently confused by students. Understanding the difference is important.
A term is a unit in an algebraic expression that is separated from other units by addition or subtraction.
A factor is one of the quantities multiplied together to make up a single term.
In 3x + 5x:
- There are two terms: 3x and 5x.
- In each term, 3 and x are factors; 5 and x are factors.
- The terms 3x and 5x are like terms and can be combined: 3x + 5x = 8x.
In 3 × x:
- This is one term: 3x.
- 3 and x are the two factors within that single term.
Combining terms involves addition and subtraction. Factors involve multiplication within a term. The article [What Is a Factor?] explains factors clearly and provides helpful context for distinguishing them from terms.
Combining Like Terms and Rational Numbers
Rational numbers are numbers that can be written as a fraction p/q where q ≠ 0. They frequently appear as coefficients of algebraic terms.
When combining like terms with rational coefficients, apply the same rule: add or subtract the rational coefficients and keep the variable unchanged.
Example: 1/2x + 3/2x
Coefficients: 1/2 + 3/2 = 4/2 = 2
Result: 2x
Example: 3/4a – 1/4a
Coefficients: 3/4 – 1/4 = 2/4 = 1/2
Result: 1/2a
The article [What Is a Rational Number?] explains rational numbers in detail, which is helpful when working with fractional coefficients in algebra.
Combining Like Terms and Real Numbers
Coefficients can be any real number. This includes positive integers, negative integers, fractions, decimals, and even irrational numbers such as √2, though these are rare in introductory algebra.
The type of real number used as a coefficient does not change the core rule. If two terms have the same variable part, their coefficients are combined regardless of what type of numbers those coefficients are.
The article [What Is a Real Number?] provides a full explanation of the real number system, which helps you understand the complete range of values that coefficients can take.
Combining Like Terms in Physics Formulas
Algebraic simplification by combining like terms is not limited to pure mathematics. In physics, formulas often contain multiple terms with the same variable, and combining them correctly simplifies calculations.
For example, when working with equations of motion or force problems, you might encounter expressions with several terms involving the same variable representing velocity, time, or force. Recognising and combining like terms in those expressions leads to a cleaner form of the equation that is easier to evaluate.
The article [What Is Force in Physics?] explains how algebraic expressions involving physical quantities such as force, mass, and acceleration are constructed, which provides useful context for understanding why algebraic simplification skills are directly applicable in scientific settings.
Worked Examples
Example 1: Two Positive Terms
Given: 4x + 6x
Identify Like Terms: Both contain x.
Combine: 4 + 6 = 10
Answer: 10x
Check: Correct.
Example 2: Positive and Negative Terms
Given: 9a – 4a
Identify Like Terms: Both contain a.
Combine: 9 – 4 = 5
Answer: 5a
Check: Correct.
Example 3: Three Like Terms
Given: 2x + 5x + 3x
Identify Like Terms: All contain x.
Combine: 2 + 5 + 3 = 10
Answer: 10x
Check: Correct.
Example 4: Four Like Terms
Given: 8m – 3m + 2m – m
Combine: 8 – 3 + 2 – 1 = 6
Answer: 6m
Check: Correct.
Example 5: Multiple Variables
Given: 3x + 4y + 2x – y
Group: (3x + 2x) + (4y – y)
Combine: 5x + 3y
Answer: 5x + 3y
Check: Both groups combined correctly.
Example 6: Squared Variables
Given: 6x² – 2x² + x²
Combine: 6 – 2 + 1 = 5
Answer: 5x²
Check: Correct.
Example 7: Cubed Variables
Given: 4a³ + 3a³ – a³
Combine: 4 + 3 – 1 = 6
Answer: 6a³
Check: Correct.
Example 8: Negative Coefficients
Given: -5x + 8x – 2x
Combine: -5 + 8 – 2 = 1
Answer: x (coefficient of 1 is not written)
Check: Correct.
Example 9: Fractions
Given: 1/3y + 2/3y
Combine: 1/3 + 2/3 = 3/3 = 1
Answer: y
Check: Correct.
Example 10: Decimal Coefficients
Given: 1.4k + 2.6k
Combine: 1.4 + 2.6 = 4
Answer: 4k
Check: Correct.
Example 11: Constants
Given: 7 + (-3) + 5
Combine: 7 – 3 + 5 = 9
Answer: 9
Check: Correct.
Example 12: Mixed Like and Unlike Terms
Given: 5a + 3b – 2a + b + 4
Group: (5a – 2a) + (3b + b) + 4
Combine: 3a + 4b + 4
Answer: 3a + 4b + 4
Check: 3a, 4b, and 4 are all unlike each other. Fully simplified.
Example 13: Parentheses
Given: 3(x + 2) + 2x
Distribute: 3x + 6 + 2x
Group: (3x + 2x) + 6
Combine: 5x + 6
Answer: 5x + 6
Check: Correct.
Example 14: Quadratic Expression
Given: 4x² + 3x – x² + 2x
Group: (4x² – x²) + (3x + 2x)
Combine: 3x² + 5x
Answer: 3x² + 5x
Check: x² and x are unlike. Fully simplified.
Example 15: Linear Expression
Given: 6x + 4 – 2x + 9
Group: (6x – 2x) + (4 + 9)
Combine: 4x + 13
Answer: 4x + 13
Check: 4x and 13 are unlike. Fully simplified.
Example 16: Multiple Variable Expression
Given: 2xy + 5x – 3xy + x
Group: (2xy – 3xy) + (5x + x)
Combine: -xy + 6x
Answer: -xy + 6x
Check: -xy and 6x are unlike (xy versus x). Fully simplified.
Example 17: Zero Coefficient
Given: 5x – 5x + 4y
Combine: 5x – 5x = 0
Answer: 4y
Check: The x terms cancel. Only 4y remains.
Example 18: Word-Based Example
Given: “The sum of three times a number and four times the same number”
Expression: 3x + 4x
Combine: 7x
Answer: 7x
Example 19: More Advanced Expression
Given: 4p² + 3pq – p² + 2pq – 5p
Group: (4p² – p²) + (3pq + 2pq) + (-5p)
Combine: 3p² + 5pq – 5p
Answer: 3p² + 5pq – 5p
Check: 3p², 5pq, and -5p are all unlike. Fully simplified.
Example 20: Checking Example
Given: 7a + 3b – 4a – 2b + 1
Group: (7a – 4a) + (3b – 2b) + 1
Combine: 3a + b + 1
Answer: 3a + b + 1
Check: 3a, b, and 1 are all unlike each other. Fully simplified.
Combining Like Terms With Parentheses
When an expression contains parentheses, apply the distributive property first to remove the brackets, then combine like terms.
Example 1: 3(x + 2) + 2x
Step 1: Distribute 3 across the bracket.
3x + 6 + 2x
Step 2: Group like terms.
(3x + 2x) + 6
Step 3: Combine.
5x + 6
Example 2: 5a + 2(a – 3)
Step 1: Distribute 2.
5a + 2a – 6
Step 2: Group.
(5a + 2a) – 6
Step 3: Combine.
7a – 6
Example 3: 4x – (2x + 5)
Step 1: Distribute the negative sign.
4x – 2x – 5
Step 2: Combine.
2x – 5
Always distribute before combining. Attempting to combine before distributing leads to errors.
Common Mistakes Students Make
| Mistake | Incorrect | Correct |
|---|---|---|
| Combining different variables | 3x + 4y = 7xy | 3x + 4y stays as 3x + 4y |
| Combining different exponents | 5x + 3x² = 8x³ | 5x + 3x² stays as 5x + 3x² |
| Adding exponents | 4x² + 4x² = 4x⁴ | 4x² + 4x² = 8x² |
| Losing the negative sign | 5x – 8x = 13x | 5x – 8x = -3x |
| Combining constant with variable term | 3x + 4 = 7x | 3x + 4 stays as 3x + 4 |
| Assuming equal coefficients mean like | 5x and 5y are like | 5x and 5y are unlike (different variables) |
| Treating factors as separate terms | In 6x, 6 and x are separate terms | In 6x, one term with factors 6 and x |
| Forgetting to distribute negative sign | 4x – (2x + 5) = 4x – 2x + 5 | 4x – 2x – 5 = 2x – 5 |
| Fraction arithmetic errors | 1/2x + 1/3x = 2/5x | LCD = 6: 3/6x + 2/6x = 5/6x |
| Changing exponent during combining | 3a² – a² = 2a | 3a² – a² = 2a² |
Combining Like Terms Rules Cheat Sheet
| Rule | Example | Result |
|---|---|---|
| Same variable → may be like terms | 4x and 7x | Like terms |
| Same exponent → required | 4x² and 7x² | Like terms |
| Add/subtract coefficients only | 4x + 7x | 11x |
| Keep variable part unchanged | 3a² + 5a² | 8a² (not 8a⁴) |
| Unlike terms stay separate | 3x + 5y | 3x + 5y |
| Constants combine with constants | 5 + (-3) | 2 |
| Negative signs stay attached | -4a + 9a | 5a |
| Distribute before combining | 2(x + 3) + x | 2x + 6 + x = 3x + 6 |
| Three or more like terms | 2x + 5x + x | 8x |
| Zero result | 5x – 5x | 0 (term disappears) |
Like Terms vs Unlike Terms
| Feature | Like Terms | Unlike Terms |
|---|---|---|
| Definition | Same variable and exponent | Different variable or exponent |
| Variables | Identical | At least one differs |
| Exponents | Identical | At least one differs |
| Can be combined? | Yes | No |
| Coefficients | May differ | Irrelevant to classification |
| Example | 4x² and -3x² | 4x² and -3x³ |
| After combining | One single term | Two or more separate terms |
Combining Like Terms vs Solving an Equation
These two processes are different and serve different purposes.
Combining like terms simplifies an expression.
3x + 5x = 8x
You are not finding what x equals. You are simply writing the expression in a more compact form.
Solving an equation finds the specific value of a variable.
3x + 5 = 20
3x = 15
x = 5
Here you use inverse operations to isolate x and find its numerical value.
Combining like terms is often a step within the process of solving an equation, but the two activities are not the same. An expression has no equals sign and no unique solution. An equation has an equals sign and can be solved.
Practice Questions
20 Multiple Choice Questions
Question 1: What is 4x + 9x?
A) 13x²
B) 36x
C) 13x
D) 4x + 9
Correct Answer: C
Explanation: Add the coefficients: 4 + 9 = 13. The variable x stays. Result: 13x.
Question 2: Simplify 8a – 3a.
A) 5a²
B) 24a
C) 11a
D) 5a
Correct Answer: D
Explanation: 8 – 3 = 5. Variable a unchanged. Result: 5a.
Question 3: What is -4x + 9x?
A) -5x
B) 5x
C) -13x
D) 13x
Correct Answer: B
Explanation: -4 + 9 = 5. Result: 5x.
Question 4: Can 3x and 5x² be combined?
A) Yes: 8x³
B) Yes: 8x²
C) No, they have different exponents
D) Yes: 15x³
Correct Answer: C
Explanation: 3x has exponent 1; 5x² has exponent 2. Different exponents make them unlike terms.
Question 5: Simplify 5a + 3b – 2a + b.
A) 7ab
B) 3a + 4b
C) 3a – 4b
D) 7a + 4b
Correct Answer: B
Explanation: (5a – 2a) + (3b + b) = 3a + 4b.
Question 6: What is 1/2x + 3/4x?
A) 4/6x
B) 5/4x
C) 3/8x
D) 5/6x
Correct Answer: B
Explanation: Convert to common denominator: 2/4x + 3/4x = 5/4x.
Question 7: Simplify 6y² – 4y² + y².
A) 3y⁶
B) 3y²
C) 2y²
D) 9y²
Correct Answer: B
Explanation: 6 – 4 + 1 = 3. Result: 3y².
Question 8: What is 7m – 7m?
A) 14m
B) 1
C) 0
D) 49m²
Correct Answer: C
Explanation: 7 – 7 = 0. The term completely disappears.
Question 9: Which terms can be combined in 3x + 4y + 2x – y?
A) 3x and 4y
B) 3x and 2x, then 4y and -y
C) 4y and 2x
D) All four terms
Correct Answer: B
Explanation: 3x and 2x are like terms; 4y and -y are like terms.
Question 10: Simplify 0.6p + 1.4p.
A) 0.8p
B) 2p
C) 2.4p
D) 0.84p
Correct Answer: B
Explanation: 0.6 + 1.4 = 2.0. Result: 2p.
Question 11: What is 3xy + 5xy?
A) 8x²y²
B) 15xy
C) 8xy
D) 8x + y
Correct Answer: C
Explanation: Both contain the variable part xy. 3 + 5 = 8. Result: 8xy.
Question 12: Can 4ab and 4a be combined?
A) Yes: 8a²b
B) Yes: 8ab
C) No: different variable parts
D) Yes: 4a(b + 1)
Correct Answer: C
Explanation: 4ab contains ab. 4a contains only a. They are unlike terms and cannot be combined.
Question 13: Simplify 3(x + 4) + 2x.
A) 5x + 12
B) 5x + 4
C) 3x + 12
D) 6x + 4
Correct Answer: A
Explanation: Distribute: 3x + 12 + 2x = 5x + 12.
Question 14: What is -3a – 5a?
A) 2a
B) -8a
C) 8a
D) -2a
Correct Answer: B
Explanation: -3 + (-5) = -8. Result: -8a.
Question 15: Simplify 4x² + 3x + 2x².
A) 9x²
B) 6x² + 3x
C) 9x⁴ + 3x
D) 6x + 3x = 9x
Correct Answer: B
Explanation: 4x² + 2x² = 6x². The term 3x is unlike x² and stays. Result: 6x² + 3x.
Question 16: Simplify 2m + 5n – m + 3n.
A) m + 8n
B) 9mn
C) 3m + 2n
D) m – 8n
Correct Answer: A
Explanation: (2m – m) + (5n + 3n) = m + 8n.
Question 17: What is 5 + (-3) + 8?
A) 0
B) -6
C) 10
D) 16
Correct Answer: C
Explanation: 5 – 3 + 8 = 10. All three are constants and are like terms.
Question 18: Simplify 9k – 4k – k + 3k.
A) 8k
B) 7k
C) 6k
D) 9k
Correct Answer: B
Explanation: 9 – 4 – 1 + 3 = 7. Result: 7k.
Question 19: Which result is incorrect?
A) 5x + 3x = 8x
B) 7a – 2a = 5a
C) 4x² + x = 5x³
D) 3mn + 2mn = 5mn
Correct Answer: C
Explanation: 4x² and x have different exponents (2 and 1). They cannot be combined. 5x³ is completely wrong.
Question 20: Simplify 4(2a – 1) + 3a.
A) 11a – 4
B) 8a – 4
C) 11a – 1
D) 14a – 4
Correct Answer: A
Explanation: Distribute: 8a – 4 + 3a = 11a – 4.
15 Simplify-the-Expression Problems
Problem 1: 7x + 4x – 3x
Solution: (7 + 4 – 3)x = 8x
Problem 2: 6a + 2b – 4a + 5b
Solution: (6a – 4a) + (2b + 5b) = 2a + 7b
Problem 3: 3x² + 7x – x² + 2x
Solution: (3x² – x²) + (7x + 2x) = 2x² + 9x
Problem 4: -5m + 3m + 8m – 2m
Solution: (-5 + 3 + 8 – 2)m = 4m
Problem 5: 4xy – 2xy + xy
Solution: (4 – 2 + 1)xy = 3xy
Problem 6: 1/2a + 1/4a + 1/4a
Solution: (2/4 + 1/4 + 1/4)a = 4/4a = a
Problem 7: 3.5x – 1.5x + 0.5x
Solution: (3.5 – 1.5 + 0.5)x = 2.5x
Problem 8: 5p + 3q – 2p – q + 4
Solution: (5p – 2p) + (3q – q) + 4 = 3p + 2q + 4
Problem 9: 2(x + 3) + 5x – 1
Solution: 2x + 6 + 5x – 1 = (2x + 5x) + (6 – 1) = 7x + 5
Problem 10: 8a³ – 3a³ + a³ – 2a
Solution: (8 – 3 + 1)a³ + (-2a) = 6a³ – 2a
Problem 11: 6mn + 4m – mn – 3m
Solution: (6mn – mn) + (4m – 3m) = 5mn + m
Problem 12: 9y – y + 3 – 7
Solution: (9y – y) + (3 – 7) = 8y – 4
Problem 13: 4(x – 2) – (x + 3)
Solution: 4x – 8 – x – 3 = 3x – 11
Problem 14: 2x²y + 5x²y – 3x²y + xy
Solution: (2 + 5 – 3)x²y + xy = 4x²y + xy
Problem 15: 0.25a + 0.75a – a + 3a
Solution: (0.25 + 0.75 – 1 + 3)a = 3a
10 Identify Like Terms Questions
Q1: In 4x + 3y – 2x + 5, which terms can be combined?
Answer: 4x and -2x can be combined.
Q2: In 7a² + 3a – 2a² + 4, which terms can be combined?
Answer: 7a² and -2a² can be combined. The 3a and 4 stand alone.
Q3: In 5mn + 3m – mn + 4n, which terms can be combined?
Answer: 5mn and -mn can be combined.
Q4: In x³ + x² + x + 1, which terms can be combined?
Answer: None. All four terms are unlike each other.
Q5: In 6p + 2q – 4p + q – 3, which terms can be combined?
Answer: 6p and -4p; 2q and q. The constant -3 stands alone.
Q6: In 3ab + 2a²b – ab + 5a²b, which terms can be combined?
Answer: 3ab and -ab; 2a²b and 5a²b.
Q7: In 8x + 7y + 2 – 3x + 1, which terms can be combined?
Answer: 8x and -3x; 2 and 1.
Q8: In 4m²n + 3mn² – m²n, which terms can be combined?
Answer: 4m²n and -m²n can be combined.
Q9: In 5k + k – 3k + 2, which terms can be combined?
Answer: 5k, k, and -3k can all be combined.
Q10: In 2xy + 3x² + xy – x², which terms can be combined?
Answer: 2xy and xy; 3x² and -x².
5 Challenge Problems
Challenge 1: Simplify 3(2x + y) – 2(x – 3y) + 4y
Solution: 6x + 3y – 2x + 6y + 4y = (6x – 2x) + (3y + 6y + 4y) = 4x + 13y
Challenge 2: Simplify 4x²y – 3xy² + 2x²y – xy² + 5xy
Solution: (4x²y + 2x²y) + (-3xy² – xy²) + 5xy = 6x²y – 4xy² + 5xy
Challenge 3: Simplify 5(a + 2b) – 3(a – b) + 2(b – a)
Solution: 5a + 10b – 3a + 3b + 2b – 2a = (5a – 3a – 2a) + (10b + 3b + 2b) = 0a + 15b = 15b
Challenge 4: Simplify 1/2x² + 3/4x² – 1/4x + 3/4x
Solution: (1/2x² + 3/4x²) + (-1/4x + 3/4x) = (2/4 + 3/4)x² + (2/4)x = 5/4x² + 1/2x
Challenge 5: Simplify 2(3m² – n) + 3(m² + 2n) – 4mn
Solution: 6m² – 2n + 3m² + 6n – 4mn = (6m² + 3m²) + (-2n + 6n) – 4mn = 9m² + 4n – 4mn
Exam Tips
- Identify every term first. Before doing anything, write out a list of all the terms in the expression and include their signs.
- Ignore coefficients when checking. When deciding whether two terms are like, focus entirely on the variable parts.
- Compare variables carefully. Ensure the same letters appear in both terms.
- Check every exponent. Even if the variables match, a single different exponent makes terms unlike.
- Keep negative signs attached. A negative sign belongs to the term immediately after it. Never separate it.
- Group like terms together. Physically write them side by side before combining to reduce errors.
- Combine coefficients only. Add or subtract only the numbers. Do not touch the variable or its exponent.
- Leave unlike terms unchanged. If terms have different variable parts, leave them exactly as they are.
- Check the final expression. Scan your answer one more time to confirm there are no remaining like terms.
- Do not confuse simplifying with solving. Combining like terms produces a simpler expression. It does not produce a value for the variable unless an equation is being solved.
Quick Revision Notes
Terms: Individual parts of an algebraic expression separated by + or – signs.
Like terms: Same variables, same exponents. Coefficients may differ.
Unlike terms: Different variables or different exponents. Cannot be combined.
How to identify like terms: Compare the variable part (letters and exponents). Ignore the coefficient.
How to group like terms: Rearrange the expression so like terms appear next to each other.
How to combine coefficients: Add or subtract the coefficients. Keep the variable part exactly as it is.
Negative terms: The negative sign belongs to the coefficient. Keep it attached.
Exponents: Never change during combining. Only coefficients change.
Multiple variables: The entire variable combination must match for terms to be like.
Fractions: Combine using standard fraction addition rules.
Constants: All constants are like terms. Combine by simple arithmetic.
Common mistakes: Adding exponents, combining unlike variables, losing negative signs, combining constants with variable terms.
Combining Like Terms Cheat Sheet
| Concept | Rule | Example |
|---|---|---|
| Like terms | Same variable and exponent | 3x² and 7x² |
| Unlike terms | Different variable or exponent | 3x² and 7x³ |
| Combine | Add/subtract coefficients only | 3x + 7x = 10x |
| Variable | Never changes | 5a – 2a = 3a (not 3) |
| Exponent | Never added or changed | 4x² + 4x² = 8x² (not 8x⁴) |
| Constants | Combine with each other | 6 + (-4) = 2 |
| Negative terms | Sign is part of coefficient | -5x + 8x = 3x |
| Fractions | Combine using LCD | 1/4x + 1/2x = 3/4x |
| Unlike terms | Leave separate | 3x + 5y = 3x + 5y |
| Distribution first | Before combining | 2(x+3) + x = 2x+6+x = 3x+6 |
Frequently Asked Questions
1. What does it mean to combine like terms?
Combining like terms means adding or subtracting the coefficients of terms that share the same variable part, while leaving the variable part unchanged. The result is a simpler expression with fewer terms.
2. How do you combine like terms?
Identify the variable parts of each term. Group terms with identical variable parts. Add or subtract their coefficients. Keep the variable part as it is.
3. What are the rules for combining like terms?
The variables must be identical. The exponents on each variable must be identical. Only the coefficients are combined. The variable part stays unchanged. Unlike terms are never combined.
4. Can x and x² be combined?
No. Although both contain x, the exponents are 1 and 2 respectively. Different exponents make them unlike terms.
5. Can x and y be combined?
No. They are different variables, making any terms built from them unlike.
6. Do coefficients have to be the same?
No. Coefficients can be completely different. 3x and 7x are like terms with different coefficients. Coefficients do not determine whether terms are like.
7. How do you combine like terms with negative numbers?
Treat the negative sign as part of the coefficient. Apply standard integer arithmetic. For example, -3x + 7x = 4x, because -3 + 7 = 4.
8. How do you combine like terms with fractions?
Find the lowest common denominator, convert both fractions to that denominator, then add or subtract the numerators. The variable part remains unchanged.
9. How do you combine like terms with exponents?
The exponents must match exactly. If they do, add or subtract the coefficients. Do not change or add the exponents. 4x³ + 3x³ = 7x³, not 7x⁶.
10. Can constants be combined?
Yes. All constants are like terms with no variable part. They can always be combined: 5 + (-3) + 8 = 10.
11. Can terms with multiple variables be combined?
Yes, but the entire variable combination must match exactly. 3xy and 5xy can be combined. 3xy and 5x²y cannot, because the exponent on x differs.
12. Why can’t unlike terms be combined?
Unlike terms represent different algebraic quantities. Adding them does not produce a single meaningful term, just as adding apples and oranges does not produce a single type of fruit.
13. What is the difference between combining terms and solving an equation?
Combining like terms simplifies an expression. Solving an equation finds the value of the variable. Combining is often a step within solving, but the two processes serve different purposes.
14. How do like terms help simplify algebraic expressions?
Combining like terms reduces the number of terms in an expression, making it shorter, neater, and easier to use in further calculations.
15. What is the easiest way to identify like terms?
Cover the coefficients and compare only the variable parts. If the letters and exponents match exactly, the terms are like terms.
Summary
Knowing how to combine like terms is one of the most fundamental skills in algebra. The process is consistent and straightforward: add or subtract the coefficients of terms that share identical variable parts, and leave the variable part exactly as it is.
Like terms must have the same variables raised to the same exponents. Unlike terms have different variable parts and cannot be combined. Constants can always be combined with other constants.
Negative signs belong to the coefficient of each term and must be kept attached throughout every step. Exponents are never changed or added during the combining process.
This skill is used in simplifying algebraic expressions, solving linear equations, working with quadratic expressions, and across every area of mathematics that involves algebra.
Final Thoughts
Understanding how to combine like terms is not just about passing a specific exercise in class. It is a skill that runs through almost every topic in algebra and beyond.
Every time you simplify an expression, solve an equation, factor a polynomial, or rearrange a formula, you are relying on the ability to correctly identify and combine like terms. Getting this right from the start makes everything that follows in mathematics easier and more reliable.
Take the time to work through the examples in this article carefully. Start with the simple two-term problems and build your way up to the mixed expressions, fractions, and multi-variable examples. Each level of practice strengthens a slightly different aspect of the skill. Together, they give you a solid and lasting foundation.
References
- OpenStax – Elementary Algebra and Prealgebra textbooks covering like terms, combining terms, and simplifying algebraic expressions.
https://openstax.org - Khan Academy – Free video lessons and exercises on combining like terms, simplifying expressions, and algebra foundations.
https://www.khanacademy.org - Mathematics LibreTexts – Open-access mathematics library covering algebraic terms, coefficients, and simplification step by step.
https://math.libretexts.org - Encyclopaedia Britannica – Reference articles on algebra, algebraic expressions, and the history of symbolic mathematics.
https://www.britannica.com - Wolfram MathWorld – Comprehensive mathematical reference covering algebraic expressions, polynomial simplification, and term combination.
https://mathworld.wolfram.com
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