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Combining Like Terms

The first real act of simplifying: gather terms that share the same variable part and add their coefficients. Learn what makes terms “like,” and why x and x^2 can never merge.

What makes terms alike

Like terms have the exact same variable part — the same letters raised to the same powers. The coefficient does not matter; the variable part is what must match. So 3x and -8x are like terms, and 5xy and 2xy are like terms, but 4x and 4x^2 are not, because x and x^2 are different objects.

The move: add the coefficients

To combine like terms, keep the variable part unchanged and add the coefficients (using signed-number arithmetic for the signs). This is really the distributive property run backwards: 3x + 5x = (3 + 5)x = 8x.

Simplify:   6a - 2b + 3 - 4a + 7b - 5

Group like terms (keep each sign):
   a-terms:  6a - 4a  = (6 - 4)a = 2a
   b-terms: -2b + 7b  = (-2 + 7)b = 5b
   constants: 3 - 5   = -2

Result:   2a + 5b - 2
Sorting into like-term groups, then adding coefficients in each group.

Care with signs

  1. Underline or colour each family of like terms first, dragging the sign in front of each term along with it.
  2. Add the coefficients of each family separately; do not let an a-term wander into the b pile.
  3. Write the result with the variable parts unchanged. The result is an equivalent expression — shorter, but worth the same for every value of the variables.