English:Combining Like Terms
Combining Like Terms
Introduction
Combining like terms is a way to simplify an algebraic expression without changing its value. When terms have exactly the same variable part, including the same exponents, you can add or subtract their coefficients. For example, 3x + 5x = 8x because three groups of x plus five groups of x make eight groups of x.
This aiMOOC is designed for learners in Grades 7–8. You will learn how to identify terms, coefficients, variables, constants, and exponents; decide which terms are like; combine positive and negative coefficients; simplify expressions with several variables; use the distributive property before combining terms; and check whether two expressions are equivalent.
The Building Blocks of an Expression
An expression is a mathematical phrase made from numbers, variables, operation signs, and sometimes exponents. In the expression 7x - 3y + 5 - 2x, the terms are 7x, -3y, 5, and -2x. A plus or minus sign belongs with the term that follows it, so the last term is -2x, not 2x.
A variable is a letter or symbol that can represent a number. A coefficient is the numerical factor multiplying a variable part. In -6a, the coefficient is -6. A constant is a term without a variable, such as 9 or -4. An exponent tells how many times a base is used as a factor, so x² means x · x.
Algebra tiles can help you see these roles. Unit tiles represent constants, long rectangular tiles can represent x, and square tiles can represent x². Positive and negative versions also help model addition and subtraction. Because the shapes for x and x² are different, the model makes it easier to see why x-terms and x²-terms are not like terms.
What Makes Terms Like?
Like terms have the same variable part with the same exponents. Their coefficients may be different.
For example, 4x and -7x are like terms. The terms 3x² and 10x² are also like terms. The terms 5ab and -2ab are like because both contain exactly ab. However, x and x² are not like, 2a and 2b are not like, and 4xy and 4x are not like. All constants are like terms with other constants because they have no variable part.
A useful test is to ignore the coefficient for a moment and compare only the variable part. If the letters and their exponents match exactly, the terms are like.
Why Combining Like Terms Works
Combining like terms is based on the distributive property. The expression 3x + 5x can be rewritten as (3 + 5)x, which equals 8x. You are adding the coefficients while keeping the common factor x.
The same idea works with subtraction: 9y - 4y = (9 - 4)y = 5y. It also works with negative coefficients: -3a + 8a = (-3 + 8)a = 5a.
The balance animation connects algebraic operations with the idea of preserving equality. Combining like terms does not change the value of an expression; it rewrites the expression in an equivalent, simpler form.
A Reliable Four-Step Method
- Identify the terms: Treat subtraction as adding a negative so that each sign stays attached to its term.
- Group like terms: Match terms only when their variable parts and exponents are identical.
- Combine coefficients: Add or subtract the numerical coefficients of each group.
- Write the result: Keep the common variable part and combine any constants.
For example, simplify 7x + 4 - 2x + 9. The x-terms are 7x and -2x, so they combine to 5x. The constants 4 and 9 combine to 13. The simplified expression is 5x + 13.
Negative Coefficients and Subtraction
Negative signs cause many errors, so keep each sign with its term. In 6m - 9m + 4, the coefficient of the second m-term is -9. Therefore 6m - 9m = -3m, and the expression simplifies to -3m + 4.
When you see a term with no written coefficient, remember that x means 1x and -x means -1x. Thus 5x - x = 5x - 1x = 4x.
A good habit is to rewrite an expression with spaces between terms and circle or mark the sign together with the term. This reduces the chance of changing a subtraction sign by accident.
Several Variables and Exponents
Expressions may contain several groups of like terms. Consider 4x + 2y - 3x + 7y. The x-terms combine to x, and the y-terms combine to 9y, so the result is x + 9y.
Exponents must match. In 6m² + 3m - 2m² + 5, the m²-terms combine to 4m², but 3m cannot combine with 4m². The constant 5 also remains separate. The simplified expression is 4m² + 3m + 5.
In an algebra-tile model, x², x, and unit tiles have different shapes. That visual difference reinforces an important rule: terms from different variable or exponent groups stay separate when you simplify.
Distribute First, Then Combine
Sometimes parentheses must be removed before all like terms are visible. Use the distributive property first.
Example: 3(2x + 5) - 4x becomes 6x + 15 - 4x. Now 6x and -4x are like terms, so the expression simplifies to 2x + 15.
A negative sign before parentheses also distributes. For example, -(x - 3) + 4x becomes -x + 3 + 4x, which simplifies to 3x + 3.
Checking Equivalent Expressions
You can check your simplification by substituting a value for the variable into both the original and simplified expressions. This does not prove every possible step by itself, but it is a useful error check.
For example, compare 5x + 2x - 3 and 7x - 3 at x = 2. The original gives 10 + 4 - 3 = 11. The simplified expression gives 14 - 3 = 11. The matching result supports that the simplification is correct.
An equivalent expression may look different while having the same value for every allowed value of its variables. Combining like terms is one of the main tools for creating such equivalent expressions.
Common Mistakes and How to Avoid Them
Mistake 1: Combining unlike variables. The expression 3x + 4y cannot become 7xy or 7x because x and y are different variable parts.
Mistake 2: Ignoring exponents. The terms 2x and 5x² are unlike because x and x² are different variable parts.
Mistake 3: Losing a negative sign. In 8a - 11a, the second coefficient is -11, so the result is -3a.
Mistake 4: Changing the variable part. When you combine 2ab + 7ab, add only the coefficients: the result is 9ab, not 9a²b².
Mistake 5: Combining before distributing. In 2(x + 3) + x, first write 2x + 6 + x, then combine to get 3x + 6.
A quick self-check is to ask: Did I combine only matching variable parts? Did I keep each sign with its coefficient? Did I leave the variable part unchanged? Did I combine all constants?
A Real-World Model
Suppose one notebook costs x dollars. If you buy 3 notebooks in one purchase and 5 more notebooks later, the total notebook cost is 3x + 5x = 8x. If you also pay a fixed delivery fee of 4 dollars, the total is 8x + 4. The constant 4 cannot combine with 8x because it represents a different kind of quantity.
This kind of reasoning appears in formulas, budgeting, measurement, science, coding, and later work with polynomials and linear equations. Simplifying first often makes a longer problem easier to understand and solve.
Interactive Tasks
Quiz: Test Your Knowledge
Which pair consists of like terms? (4x and negative 2x) (!4x and 4y) (!3x and 3x squared) (!5ab and 5a)
Simplify 3x plus 5x. (8x) (!8) (!15x) (!2x)
Simplify 7a plus 2 minus 4a plus 5. (3a plus 7) (!11a plus 7) (!3a plus 3) (!7a plus 3)
Why are x and x squared not like terms? (Their exponents are different) (!Their coefficients are hidden) (!They use the same variable) (!They are both positive)
Simplify negative 6y plus 2y. (Negative 4y) (!Negative 8y) (!4y) (!Negative 4)
Simplify 2m plus 3n plus m. (3m plus 3n) (!6mn) (!5m plus n) (!3m plus n)
Simplify 4 times the quantity 2x plus 1, then subtract 3x. (5x plus 4) (!5x plus 1) (!11x plus 4) (!8x minus 2)
What is the result of combining the constants 9 and negative 4? (5) (!13) (!Negative 5) (!36)
What is the coefficient of p in negative 7p? (Negative 7) (!7) (!p) (!Negative p)
Which expression is equivalent to 5x plus 2x minus 3? (7x minus 3) (!7x) (!7x minus 1) (!10x minus 3)
Memory Game
| Coefficient | Numerical factor multiplying a variable part |
| Variable | Symbol that can represent a number |
| Constant | Term without a variable |
| Like terms | Terms with identical variable parts and exponents |
| Distributive property | Rule used to remove parentheses by multiplication |
| Equivalent expression | Different-looking form with the same value |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Eight x | Three x plus five x |
| Negative two y | Four y minus six y |
| Seven a plus three | Two a plus five a plus three |
| Four m squared | Nine m squared minus five m squared |
| Three p plus six | Two times p plus three, then add p |
...
Crossword Puzzle
| Coefficient | What is the numerical factor multiplying a variable part called? |
| Variable | What symbol can represent an unknown or changing number? |
| Constant | What term contains no variable? |
| Exponent | What indicates the power of a variable? |
| Expression | What mathematical phrase contains terms but no required equality sign? |
| Distributive | What property lets multiplication act on every term inside parentheses? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Like terms sort: Create twelve algebra cards and sort them into groups of like terms; explain one correct match and one pair that must stay separate.
- Coefficient hunt: Write eight terms with positive, negative, and unwritten coefficients, then label every coefficient and variable part.
- Algebra tiles model: Use paper or digital algebra tiles to model an expression such as 3x + 2x + 4 and show visually why the x-terms combine but the constant stays separate.
- Error detective: Invent three common mistakes in combining like terms and write a corrected solution with a short explanation for each one.
Standard
- Simplification poster: Design a one-page poster that teaches the four-step method for combining like terms and includes at least four worked examples.
- Math interview: Interview a classmate about how they decide whether two terms are like, record the explanation, and compare it with the exact variable-and-exponent rule.
- Expression video: Record a short teaching video in which you simplify an expression containing negative coefficients and explain every sign decision.
- Real-world algebra model: Create a shopping, distance, or measurement situation that leads to an expression with at least two groups of like terms, then simplify and interpret the result.
Advanced
- Distribute and combine investigation: Create six expressions with parentheses, simplify them by distributing and combining like terms, and explain why changing the order of valid steps does not change the final equivalent expression.
- Equivalence experiment: Choose three original-and-simplified expression pairs, test each pair with at least five substitution values, and analyze what the matching outputs suggest about equivalence.
- Algebra misconception study: Collect anonymous examples of mistakes from practice work, classify the errors by type, and design a mini-lesson targeting the most frequent misconception.
- Expression design challenge: Build three different unsimplified expressions that all simplify to the same target expression, then justify algebraically why each construction works.
Learning Assessment
- Reasoning with signs: Simplify an expression with at least four terms and two negative coefficients, then explain how each sign affected the coefficient calculation.
- Comparing expressions: Decide whether two multi-term expressions are equivalent after simplification and justify your conclusion with algebraic steps and a substitution check.
- Unlike-term explanation: Explain why 4x, 4x², and 4y cannot all be combined even though they share some symbols or numbers.
- Distributive transfer: Simplify an expression containing parentheses and like terms, then identify exactly where the distributive property was required.
- Model and interpret: Translate a real-world situation into an algebraic expression, combine like terms, and explain what each term in the simplified expression represents.
- Create and verify: Write an unsimplified expression that simplifies to 5a - 2b + 7, and verify your result by simplifying and substituting sample values.
Evidence of Learning
Knowledge: You can define term, coefficient, variable, constant, exponent, like terms, equivalent expression, and distributive property, and you can state the exact rule for identifying like terms.
Skills: You can preserve signs, group matching variable parts, combine coefficients, simplify constants, distribute before combining when needed, and check results by substitution.
Products: Your worked solutions, card sorts, posters, algebra-tile models, videos, investigations, and created expressions show how you represent and communicate algebraic reasoning.
Reasoning: You can explain why combining coefficients is valid and why unlike variable parts or different exponents must remain separate.
Transfer: You can use combining like terms in unfamiliar contexts such as formulas, budgeting, measurement, polynomial work, and solving equations.
OERs on the Topic
Linked Learning Areas