Combining Like Terms (cont)
Once an expression has four or more terms — something like — combining like terms becomes a sorting job. Group the terms that match, combine each group, and write the shortened result. The math is the same as before; the new challenge is keeping track of signs while you sort.
The rule that makes it all work: the sign in front of a term travels with that term. In , the third term is , not . Treat every term as owning its sign and the sorting becomes safe.
Group, then combine
Scan the expression and sort the terms into groups: all the terms together, all the constants together, each term carrying its own sign. For the groups are and .
Combine each group separately: and . Write the groups back together: . Two groups in, two terms out — the expression cannot get any shorter, because and are not like terms.
Watch the signs
Expressions that start with a negative term, like , punish careless sorting. The terms are and , which combine to . The constants are and , which combine to . Result: .
If sign-juggling trips you up, underline each term with its sign before you sort. It feels slow for a week and then saves you points forever.
When the variable terms cancel
Sometimes a group combines to zero. In , the terms give , and the term vanishes. All that remains is . An answer with no variable in it is completely legitimate — do not go hunting for an that is no longer there.
Worked examples
Example 1: two groups
Simplify .
Answer:
Example 2: negatives in front
Simplify .
Answer:
Example 3: the variable terms cancel
Simplify .
Answer:
Try one yourself
Common questions
How do I know which sign belongs to which term?
The sign immediately to the left of a term belongs to it. In , read the terms as , , , . If a term starts the expression with no sign, it is positive.
Does the order of my answer matter?
Not mathematically — and are equal. The convention is to write the variable term first, and matching the convention makes your answer easier to check against a key.
What if there are three kinds of terms, like 's, 's, and constants?
Make three groups. For , combine , then , then the lone constant stays. The answer is .
Why can't I combine and at the end?
They are not like terms — one has a variable and one does not. Ten of something unknown plus a plain ten are different quantities. is the finished answer.
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