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➕ Algebra 1 · Equations

Combining Like Terms: notes and common mistakes

Study notes

Key ideas

  • A term is one chunk of an expression, separated by plus and minus signs. Each term keeps the sign in front of it.
  • Like terms have exactly the same variable part: the same letters with the same exponents. The coefficients may differ. Constants are like terms with each other.
  • To combine like terms, add the coefficients and keep the variable part: ax + bx = (a + b)x.
  • Unlike terms stay separate. The simplified expression has the same value as the original for every value of the variable.

Steps

  1. Mark each term together with the sign in front of it.
  2. Group the like terms (same variable part), including the constants.
  3. Add the coefficients in each group. A lone letter has coefficient 1, and −x has coefficient −1.
  4. Write the result, usually with the highest power first and the constant last.
  5. Check with a test number such as 3 or 5 in the original and the result.

Formulas

  • Combine rule: ax + bx = (a + b)x
  • Minus before a group: −(a + b) = −a − b and −(a − b) = −a + b
  • Cancelling: ax − ax = 0

Watch out

  • A lone letter has coefficient 1: 9b − b = 8b, not 9.
  • Only the coefficients are added. 3x + 5x = 8x, never 8x2. Adding never changes the exponent.
  • x and x2 are not like terms. Neither are xy and x.
  • The sign belongs to the term on its right: in 9 − 4y the term is −4y.
  • A minus sign in front of parentheses flips every sign inside: −(2k − 3) = −2k + 3.

Why it works

  • Why does it work? 3x + 5x = (3 + 5)x is the distributive property read backward: 3 of something plus 5 of the same something is 8 of it.
  • Why can’t unlike terms combine? They are different quantities. At x = 3, x + x2 = 12 but 2x2 = 18, so the two expressions are not equal.
  • Why may we rearrange terms? The order of addition does not matter, so you can slide terms next to their like terms as long as each sign moves with its term.

Common mistakes

✗ Mistake6x + 5x = 11x2.
✓ InsteadAdding counts how many x’s you have, so 6x + 5x = 11x. Adding never changes the exponent. Multiplying can: 6x × 5x = 30x2.
✗ Mistake5x + 2 = 7x.
✓ Instead5x and 2 are not like terms (one has an x, the other does not), so 5x + 2 cannot be simplified further.
✗ Mistake6x + x = 6x, because the lone x has no number.
✓ InsteadA lone x has an invisible coefficient of 1, so 6x + x = 7x.
✗ Mistake−(2x − 3) = −2x − 3.
✓ InsteadThe minus sign flips every sign inside the parentheses: −(2x − 3) = −2x + 3.
✗ Mistake4x2 + 3x = 7x3 (or 7x2).
✓ Insteadx2 and x are different variable parts, so they are not like terms. 4x2 + 3x stays as it is.

Teach it back

Explain to a friend how to simplify 5n + 8 − 2n − 3 + n, and how you would check your answer with a test number.

Explaining it out loud shows what you really know. Did you…

  • I sorted the terms into n-terms and constants.
  • I remembered that the lone n has coefficient 1: 5n − 2n + n = 4n.
  • I combined the constants, 8 − 3 = 5, and wrote 4n + 5.
  • I checked with n = 3: both the original and the result equal 17.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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