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Algebra Toolkit

Algebra and Sequences

Which terms combine

  • Like terms have exactly the same letters raised to exactly the same powers, whatever their coefficients
    • 2x and 3x are like, and so are 5xy and −7xy
  • Different letters or different powers make terms unlike, so they cannot be combined
    • 2x and 3y are unlike, and so are 4x² and 6x⁴
  • Order of multiplication does not matter, so xy and yx count as like terms

Combining them

  • Add or subtract the coefficients and keep the letters unchanged
  • Each term carries the sign written in front of it, and that sign travels with the term if the order is rearranged
  • A subtraction can be read as adding a negative, which keeps the signs straight when terms are reordered
Worked example

Simplifying by collecting like terms

Simplify 7p + 4q − 2p − 9q.

Solution:

  • Like terms carry exactly the same letter to exactly the same power, and only like terms can be combined
  • Here there are two families, the p terms and the q terms
  • Take the sign in front of each term with it, so the four terms are +7p, +4q, −2p and −9q
  • The p terms: 7p − 2p = 5p
  • The q terms: 4q − 9q = −5q, which is negative because 9 is the larger number
  • The simplified expression is 5p − 5q
  • The two parts cannot be combined any further, because p and q stand for different quantities — 5p − 5q is not 0
  • Check by substituting: p = 2 and q = 1 gives 14 + 4 − 4 − 9 = 5 in the original, and 10 − 5 = 5 in the answer