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Introduction
Combining Like Terms helps you connect a mathematical rule to a result you can explain and check. Combine coefficients only when the variable factors and their powers match. Explain the result using distribution in reverse. This lesson belongs to algebra foundations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in combining like terms.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Identify coefficients and add signed numbers.
Previous lesson: Absolute Value
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- coefficient
- A factor multiplying a variable term.
- constant
- A value fixed within a problem.
- expression
- A mathematical phrase representing a value.
- factor
- An expression multiplied by another to form a product.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Combine like terms
Combine coefficients only when the variable factors and their powers match. Explain the result using distribution in reverse.
Like terms have identical variable factors raised to identical powers. Coefficients may differ. Thus 3x and −2x are like terms, but x and x² are not. Constants are like terms with no variable factor.
Combining like terms is an application of distribution: ax+bx=(a+b)x. Group variable terms and constants separately, keeping every term’s sign. A simplified expression remains equivalent to the original for every permitted variable value.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. 12x and 2x are like terms: both contain x to the first power.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Simplify 12x + 33 + 2x − 13.
- 12x and 2x are like terms: both contain x to the first power.
- Add their coefficients: (12+2)x=14x.
- Combine constants: 33−13=20. Thus the result is 14x + 20.
Answer: 14x + 20
Verify: At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
Avoid this mistake: Adding unlike powers: x+x² cannot become 2x³.
Example 2 · Basic
Simplify 13x + 33 + 2x − 14.
- 13x and 2x are like terms: both contain x to the first power.
- Add their coefficients: (13+2)x=15x.
- Combine constants: 33−14=19. Thus the result is 15x + 19.
Answer: 15x + 19
Verify: At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
Avoid this mistake: Losing subtraction signs when regrouping.
Example 3 · Intermediate
Simplify 14x + 33 + 2x − 15.
- 14x and 2x are like terms: both contain x to the first power.
- Add their coefficients: (14+2)x=16x.
- Combine constants: 33−15=18. Thus the result is 16x + 18.
Answer: 16x + 18
Verify: At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
Avoid this mistake: Multiplying coefficients while adding like terms: add the coefficients.
Example 4 · Advanced
Simplify 2x + 17 + 3x − 3.
- 2x and 3x are like terms: both contain x to the first power.
- Add their coefficients: (2+3)x=5x.
- Combine constants: 17−3=14. Thus the result is 5x + 14.
Answer: 5x + 14
Verify: At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
Avoid this mistake: Adding unlike powers: x+x² cannot become 2x³.
Example 5 · Challenge
Simplify 3x + 17 + 3x − 4.
- 3x and 3x are like terms: both contain x to the first power.
- Add their coefficients: (3+3)x=6x.
- Combine constants: 17−4=13. Thus the result is 6x + 13.
Answer: 6x + 13
Verify: At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x.
Avoid this mistake: Losing subtraction signs when regrouping.
Common mistakes and corrections
- Adding unlike powers: x+x² cannot become 2x³.
- Losing subtraction signs when regrouping.
- Multiplying coefficients while adding like terms: add the coefficients.
Real-world applications
- Combine charges for identical items on an invoice. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Add lengths expressed in terms of the same unknown measurement. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Combine coefficients only when the variable factors and their powers match. Explain the result using distribution in reverse.
Like terms have identical variable factors raised to identical powers. Coefficients may differ. Thus 3x and −2x are like terms, but x and x² are not. Constants are like terms with no variable factor.
Frequently asked questions
What should I know before combining like terms?
Identify coefficients and add signed numbers.
How can I check my work?
At x=3, the original and simplified expressions agree. Combining coefficients proves the identity for all x. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Adding unlike powers: x+x² cannot become 2x³. Losing subtraction signs when regrouping. Multiplying coefficients while adding like terms: add the coefficients.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Combine charges for identical items on an invoice.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Combine coefficients only when the variable factors and their powers match. Explain the result using distribution in reverse.
- Adding unlike powers: x+x² cannot become 2x³.
- Losing subtraction signs when regrouping.
- Multiplying coefficients while adding like terms: add the coefficients.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials