Collecting like terms

1. What are like terms?

Key idea Like terms have exactly the same letters, with the same powers. Only the number in front can be different.

3x, 8x, −x and 12x are all x terms, so they are like terms.
Numbers on their own, like 6 and −2, are like terms with each other. They are the constants.
The order of the letters does not matter: ab and ba both mean a × b. We write it as ab.

The two terms	Like terms?	Why
3x and 8x	Yes	The same letter, x.
5y and y	Yes	y means 1y. The 1 is not written.
4a and 4b	No	Different letters. Having the same number in front does not matter.
x and x²	No	x² means x × x. The powers are different.
2ab and 7ba	Yes	ba is the same as ab.
3x²y and 5xy²	No	x²y is x × x × y, but xy² is x × y × y.
6 and −2	Yes	Both are numbers on their own.
4x and 4	No	One has a letter and the other does not.

Watch out x and x² are not like terms, even though they both use x. Check the powers as well as the letters.

Example 1: sort the terms into groups

Here are some terms: 5p, 3q, p², −2p, 7, pq, 4qp, q, −1. Put the like terms together.

Group	Terms	Why
p terms	5p, −2p	The sign stays with its term.
q terms	3q, q	q means 1q.
p² terms	p²	It has no partner: p² is not a p term.
pq terms	pq, 4qp	qp is the same as pq.
numbers	7, −1	The constants.

2. The method: spot, group, combine

Key idea To collect like terms, add or subtract the numbers in front. The letters stay the same.

Why it works: 3x means three lots of x, and 2x means two lots of x. Together that is five lots of x. Each tile below is one x.

So 3x + 2x = 5x. The x does not change, because the things being counted have not changed.

Method

Spot the like terms

Each term takes the sign in front of it. In 5x − 3x, the second term is −3x.
Group them together

Move each term with its sign. You can add terms in any order.
Combine each group

Add or subtract the numbers in front. Keep the letters and powers exactly as they are.

Example 2: adding

Simplify 3x + 2x + 4x.

Spot the like terms
3x + 2x + 4x
They are all the same kind of term.
Combine each group: add or subtract the numbers in front
3x + 2x + 4x = 9x

Answer9x

3 + 2 + 4 = 9, so there are 9 lots of x.

Example 3: adding and subtracting

Simplify 6a − 3a + 4a.

Spot the like terms
6a − 3a + 4a
They are all the same kind of term.
Combine each group: add or subtract the numbers in front
6a − 3a + 4a = 7a

Answer7a

Work from left to right: 6 − 3 = 3, then 3 + 4 = 7.

Example 4: a term with no number in front

Simplify 5m + m − 2m.

Spot the like terms
5m + m − 2m
They are all m terms. m on its own means 1m.
Combine each group: add or subtract the numbers in front
5m + m − 2m = 4m

Answer4m

m means 1m, so the numbers are 5 + 1 − 2 = 4.

Example 5: when the answer is negative, or zero

Simplify (a) 4p − 7p (b) 3y − 5y + 2y.

Question	Numbers in front	Answer
(a) 4p − 7p	4 − 7 = −3	−3p
(b) 3y − 5y + 2y	3 − 5 + 2 = 0	0

−3p is a complete answer: keep the minus sign in front. When the terms cancel out, write 0, not 0y.

3. Two or more kinds of term

Key idea Collect each group on its own. Different groups stay separate, so 9a + 7b is already as simple as it can be.

Example 6: two letters

Simplify 7a + 4b + 2a + 3b.

Spot the like terms
7a + 4b + 2a + 3b
The like terms share a colour.
Group them, each with the sign in front of it
7a + 2a + 4b + 3b
Combine each group: add or subtract the numbers in front
7a + 2a = 9a
4b + 3b = 7b

Answer9a + 7b

Example 7: two letters, with subtraction

Simplify 5x + 4y − 3x + 5y.

Spot the like terms
5x + 4y − 3x + 5y
The like terms share a colour.
Group them, each with the sign in front of it
5x − 3x + 4y + 5y
Combine each group: add or subtract the numbers in front
5x − 3x = 2x
4y + 5y = 9y

Answer2x + 9y

Watch out The sign in front belongs to the term. In 5x + 4y − 3x + 5y it is −3x that moves next to 5x, minus sign and all.

Example 8: numbers on their own as well

Simplify 2c − 5d + 4 + 3c + 7d + 2.

Spot the like terms
2c − 5d + 4 + 3c + 7d + 2
The like terms share a colour.
Group them, each with the sign in front of it
2c + 3c − 5d + 7d + 4 + 2
Combine each group: add or subtract the numbers in front
2c + 3c = 5c
−5d + 7d = 2d
4 + 2 = 6

Answer5c + 2d + 6

The numbers on their own are a group too. They usually go last.

Example 9: starting with a negative term

Simplify −6x + 5y + 4 + 2x − 3y − 7.

Spot the like terms
−6x + 5y + 4 + 2x − 3y − 7
The like terms share a colour.
Group them, each with the sign in front of it
−6x + 2x + 5y − 3y + 4 − 7
The first term is −6x: the minus sign belongs to it.
Combine each group: add or subtract the numbers in front
−6x + 2x = −4x
5y − 3y = 2y
4 − 7 = −3

Answer−4x + 2y − 3

Example 10: a group that cancels out

Simplify 3a − 2b + 5 − 3a + b.

Spot the like terms
3a − 2b + 5 − 3a + b
The like terms share a colour.
Group them, each with the sign in front of it
3a − 3a − 2b + b + 5
Combine each group: add or subtract the numbers in front
3a − 3a = 0
−2b + b = −b
5 has nothing to collect with

Answer−b + 5

The a terms make 0, so they disappear. −b + 5 can also be written 5 − b. Both are right.

Watch out 9a + 7b is not 16ab. Unlike terms can’t be added together. Test it with numbers: if a = 1 and b = 10, then 9a + 7b = 9 + 70 = 79, but 16ab = 160.

4. Powers, and terms with more than one letter

Key idea Check the letters and the powers. x², x and x³ are three different groups. Adding never changes the power: 3x² + 2x² = 5x².

Example 11: squares and plain x terms

Simplify 8x² + 5x − 4x² + 7x.

Spot the like terms
8x² + 5x − 4x² + 7x
x² terms and x terms are different groups.
Group them, each with the sign in front of it
8x² − 4x² + 5x + 7x
Combine each group: add or subtract the numbers in front
8x² − 4x² = 4x²
5x + 7x = 12x

Answer4x² + 12x

Example 12: x²y terms and x² terms

Simplify 6x²y − 2x² + 4x²y − 5x².

Spot the like terms
6x²y − 2x² + 4x²y − 5x²
The like terms share a colour.
Group them, each with the sign in front of it
6x²y + 4x²y − 2x² − 5x²
Combine each group: add or subtract the numbers in front
6x²y + 4x²y = 10x²y
−2x² − 5x² = −7x²

Answer10x²y − 7x²

x²y has a y in it and x² does not, so they are different groups.

Example 13: the letters in a different order

Simplify 3ab + 5ba − 2ab.

Spot the like terms
3ab + 5ba − 2ab
ab and ba are like terms.
Combine each group: add or subtract the numbers in front
3ab + 5ba − 2ab = 6ab

Answer6ab

ba means b × a, which is the same as a × b. Write the answer with the letters in alphabetical order.

Example 14: harder: x²y and xy²

Simplify 4xy² + 2x²y − xy² + 3x²y.

Spot the like terms
4xy² + 2x²y − xy² + 3x²y
The like terms share a colour.
Group them, each with the sign in front of it
4xy² − xy² + 2x²y + 3x²y
Combine each group: add or subtract the numbers in front
4xy² − xy² = 3xy²
2x²y + 3x²y = 5x²y

Answer3xy² + 5x²y

In xy² the y is squared. In x²y the x is squared. They are not like terms.

Example 15: a power that cancels out

Simplify 5y² − 3y + 2 − 5y² + 8y.

Spot the like terms
5y² − 3y + 2 − 5y² + 8y
The like terms share a colour.
Group them, each with the sign in front of it
5y² − 5y² − 3y + 8y + 2
Combine each group: add or subtract the numbers in front
5y² − 5y² = 0
−3y + 8y = 5y
2 has nothing to collect with

Answer5y + 2

Watch out 3x + 2x = 5x, not 5x². You are counting x’s, not multiplying them. And 4x² + 3x stays as it is: it is not 7x² or 7x³.

This lesson is about adding and subtracting terms. Multiplying terms, such as 3a × 4b or x² × x³, follows different rules and is not covered here.

5. Fractions and decimals in front

Key idea The method is the same. Add and subtract the numbers in front, using what you know about decimals and fractions.

Example 16: decimals

Simplify 2.5a + 1.3a − 0.8a.

Spot the like terms
2.5a + 1.3a − 0.8a
They are all the same kind of term.
Combine each group: add or subtract the numbers in front
2.5a + 1.3a − 0.8a = 3a

Answer3a

Example 17: decimals and numbers on their own

Simplify 1.2x + 3 − 0.5x − 4.5.

Spot the like terms
1.2x + 3 − 0.5x − 4.5
The like terms share a colour.
Group them, each with the sign in front of it
1.2x − 0.5x + 3 − 4.5
Combine each group: add or subtract the numbers in front
1.2x − 0.5x = 0.7x
3 − 4.5 = −1.5

Answer0.7x − 1.5

Example 18: fractions

Simplify 12x + 14x.

Spot the like terms
12x + 14x
They are all the same kind of term.
Combine each group: add or subtract the numbers in front
12x + 14x = 34x

Answer34x

12 = 24, so 24 + 14 = 34.

Example 19: a letter over a number

Simplify x3 + x3 + y − y4.

Spot the like terms
x3 + x3 + y − y4
The like terms share a colour.
Combine each group: add or subtract the numbers in front
x3 + x3 = 2x3
y − y4 = 3y4

Answer2x3 + 3y4

x3 is the same as 13x, so the numbers are 13 + 13 = 23. y is 1y, and 1 − 14 = 34. You can also write the answer as 23x + 34y.

Watch out Keep fractions as fractions. Write 2x3, not 0.67x: a rounded decimal is not exact.

6. The other ways an exam asks for it

Key idea A question might not say “simplify”. A perimeter, a missing term, “show that” and an algebra pyramid all ask you to collect like terms.

Example 20: the perimeter of a rectangle

A rectangle is 3x + 2 cm long and x − 1 cm wide. Write an expression for its perimeter. Give your answer in its simplest form.

A rectangle has two lengths and two widths. Add all four sides

3x + 2 + x − 1 + 3x + 2 + x − 1
Collect like terms
3x + x + 3x + x = 8x
2 − 1 + 2 − 1 = 2

Answer8x + 2 cm

Example 21: sides that are not labelled

This pentagon is symmetrical. Write an expression for its perimeter, in its simplest form. The lengths are in centimetres.

Use the symmetry: the two walls match, and so do the two sloping edges

Five sides: the base, two walls of y + 1, and two edges of 2y + 3.
Add all five sides

4y + y + 1 + y + 1 + 2y + 3 + 2y + 3
Collect like terms
4y + y + y + 2y + 2y = 10y
1 + 1 + 3 + 3 = 8

Answer10y + 8 cm

Example 22: find the missing term

Fill in the box to make each one true.

Question	Think	Answer
4a + □ = 9a	4 and what make 9? 5	5a
7y − □ = 2y	7 take away what leaves 2? 5	5y
6a + □ − 2b = 4a − 2b	The b terms already match. 6 and what make 4? −2	−2a

Work one group at a time. The missing term can be negative.

Example 23: show that

Show that 7n + 3 − 2n − 8 + n ≡ 6n − 5.

Work on the left-hand side only

7n + 3 − 2n − 8 + n
Collect like terms, and write every step
7n − 2n + n = 6n
3 − 8 = −5
Finish by saying it matches

7n + 3 − 2n − 8 + n = 6n − 5, which is the right-hand side.

The sign ≡ means the two sides are equal for every value of n. That makes it an identity (lesson 1). In a “show that” question the answer is given, so the marks are for the working.

Example 24: an algebra pyramid

Each brick is the sum of the two bricks underneath it. Fill in the pyramid.

?

??

2xx + 34x − 1

Middle left: 2x + x + 3 = 3x + 3
Middle right: x + 3 + 4x − 1 = 5x + 2
Top: 3x + 3 + 5x + 2 = 8x + 5

8x + 5

3x + 35x + 2

2xx + 34x − 1

If a brick lower down is missing, work out what must be added, one group at a time. For example, if the top is 8x + 5 and one middle brick is 3x + 3, the other is 5x + 2: 3x and 5x make 8x, and 3 and 2 make 5.

Example 25: check your answer with a number

Mo simplifies 5x + 3 − 2x + 1 and gets 3x + 4. Check it with x = 2.

The question: 5 × 2 + 3 − 2 × 2 + 1 = 10 + 3 − 4 + 1 = 10
Mo’s answer: 3 × 2 + 4 = 6 + 4 = 10

The two match. If they had come out different, the answer would be wrong.

A matching check is good evidence, not a proof. Use a value like 2 or 3, not 0 or 1, because 0 and 1 can hide mistakes.

7. Simplify it: type your answer

Work it out on paper, then type your answer. If it is not right, you are told which group to look at again.

How it works

Type letters and numbers on the keypad or your keyboard. For a square, type the letter and then the □² key (or ^2 on a keyboard).
Two right first time moves you up a level. The levels match the collector above.

8. Practice: pick your level

A mix of everything on this page. Write your working on paper, then pick an answer. Move up a level when you are getting most of them right.

The mistakes that cost marks

Adding unlike terms. 3a + 2b written as 5ab. different letters stay separate: 3a + 2b is already simplified.
Changing the power when adding. 3x + 2x written as 5x². add the numbers and keep the letters: 5x.
Joining x² terms and x terms. 4x² + 3x written as 7x² or 7x³. different powers are different groups, so leave it as 4x² + 3x.
Joining a number to a letter term. 4x + 3 written as 7x. a number on its own is a different group: 4x + 3.
Leaving a sign behind. 5x + 2y − 3x, collected as 5x + 3x = 8x. the minus sign moves with its term: 5x − 3x = 2x.
Forgetting the hidden 1. 6y − y written as 6 or as 6y. y means 1y, so 6y − y = 5y.
Writing 1x or 0x. write x, not 1x. If a group adds up to 0, leave it out.
Treating ab and ba as different. they are like terms. Write both as ab and collect them.
Rounding a fraction. x3 + x3 written as 0.67x. keep it exact: 2x3.
Missing sides in a perimeter. Adding only the two labelled sides of a rectangle. count the sides first. Equal sides and symmetry tell you the lengths that are not labelled.

Quick checklist

Tick each one when you can do it without looking back.

I can tell whether two terms are like terms, checking the letters and the powers.
I know that ab and ba are like terms, but x²y and xy² are not.
I can collect like terms by adding and subtracting the numbers in front, keeping the letters the same.
I move each term with the sign in front of it.
I can simplify expressions with two or more letters and numbers on their own.
I write x rather than 1x, and leave out a group that adds up to 0.
I can collect terms with decimals and fractions in front.
I can write a perimeter in its simplest form, using equal sides and symmetry.
I can find a missing term, fill in an algebra pyramid and answer a “show that”.
I can check a simplified answer by putting in a number.