Collecting Like Terms

What "collecting like terms" means

In algebra a term is a single chunk of an expression, such as 4a, −3x, or 7. An expression like 4a + 3a − 2 is built from several terms joined by + and − signs.

Collecting like terms (also called combining like terms) means tidying an expression by joining together the terms that are "the same kind". It makes long expressions shorter and easier to work with, without changing their value.

If letters in maths are still new to you, read algebra basics first, then come back here.

What makes terms "like"?

Every term has two parts:

• the coefficient — the number part (in 4a the coefficient is 4),
• the letter part — the variables and their powers (in 4a the letter part is a).

Like terms have exactly the same letter part. The coefficients can be different.

Like terms (can combine)	Unlike terms (cannot combine)
3x and 8x	3x and 8y
5ab and 2ab	5ab and 2a
4y² and y²	4y² and 4y

The reason: 3x means "3 lots of x" and 8x means "8 lots of x", so together they make 11x. But 3x and 8y count different unknowns, so they stay separate — just like 3 apples and 8 bananas don't merge into one number of fruit of a single type.

The golden rule

To collect like terms, add or subtract the coefficients and keep the letter part exactly the same.

You never change the letters; you only do arithmetic on the numbers in front of them.

Worked example 1: one letter

Simplify 5x + 2x.

5x + 2x
= (5 + 2)x      (both are "lots of x", so add the coefficients)
= 7x

Why it works: five x's and two more x's give seven x's. The x is just a label that comes along for the ride.

Worked example 2: subtraction

Simplify 9m − 4m.

9m − 4m
= (9 − 4)m
= 5m

Worked example 3: keeping the signs

Each + or − belongs to the term after it. Treat − as a negative coefficient.

Simplify 7y − 10y.

7y − 10y
= (7 − 10)y
= −3y

A negative answer is perfectly fine. −3y simply means "3 lots of x in the negative direction".

Worked example 4: numbers and letters mixed

When an expression has both letter terms and plain numbers, collect each kind separately.

Simplify 6a + 5 − 2a + 3.

Step 1 — underline or label the like terms:

6a  +5  −2a  +3
↑ a-terms: 6a and −2a
↑ number terms: +5 and +3

Step 2 — collect the a-terms: 6a − 2a = 4a. Step 3 — collect the numbers: 5 + 3 = 8. Step 4 — write the simplified expression:

6a + 5 − 2a + 3 = 4a + 8

By convention we write the letter term first, then the number.

Worked example 5: more than one letter

Simplify 3x + 4y + 2x − y.

Group the x-terms and the y-terms:

x-terms:  3x + 2x = 5x
y-terms:  4y − y  = 3y     (remember −y means −1y)

So 3x + 4y + 2x − y = 5x + 3y. The two groups stay separate because x and y are unlike.

Worked example 6: terms with two letters

Simplify 5ab + 2a + 3ab.

5ab and 3ab are like terms (same letter part ab). 2a is different.

5ab + 3ab = 8ab
2a stays as it is

Answer: 8ab + 2a.

A reliable method

1. Look at each term and write down its letter part.
2. Group terms that share the same letter part (watch the sign in front).
3. Add or subtract the coefficients in each group.
4. Write the answer, usually letters first (alphabetical), numbers last.

Activity: try these, then check

Simplify each expression.

1. 7c + 2c
2. 10k − 6k
3. 4x − 9x
4. 8 + 3n − 5 + n
5. 6p + 2q − p + 4q
6. 5xy + 3x + 2xy

Answers:

1. 9c (7 + 2 = 9)
2. 4k (10 − 6 = 4)
3. −5x (4 − 9 = −5)
4. 4n + 3 (3n + n = 4n; 8 − 5 = 3)
5. 5p + 6q (6p − p = 5p; 2q + 4q = 6q)
6. 7xy + 3x (5xy + 2xy = 7xy; 3x is unlike)

Where this leads

Collecting like terms is the first move in almost every algebra task. You will use it when simplifying expressions, when solving linear equations, and whenever you expand brackets and tidy the result.