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Combining ("Gathering") Like Terms

The terms of an expression are the parts of a mathematical expression that are separated by a plus (+) or minus (–) sign.

Each term is either a number or the product of a number (sometimes an understood $1$ ) and one or more variables.

For Example,

$6 x + 4 y - 2 x + 2$

The four terms of the above expression are $6 x$ , $4 y$ , $- 2 x$ and $2$ .

Like Terms:

Terms are like terms if their variable parts are the same.

For example:

$4 x + 18 y - 3 x - 7 x + 3 x y$

In the above expression $4 x$ , $- 3 x$ and $- 7 x$ are like terms. $18 y$ and $3 x y$ are unlike terms.

Like terms can be added or subtracted using the Distributive Property . The variable part stays the same; we just add the coefficients (the numerical portion of the terms). Unlike terms cannot be added or subtracted.

Example:

Simplify $12 a + 3 b - 8 b - 8 a + a - 7 b$ .

Rearrange and group like terms using the Commutative and Associative Properties.

$(12 a - 8 a + a) + (3 b - 8 b - 7 b)$

Use the Distributive Property to combine like terms.

$(12 - 8 + 1) a + (3 - 8 - 7) b$

$= 5 a - 12 b$