Collecting like terms - KS3 Maths

Key points

An image of two expressions. The first expression is four x minus two x plus eight plus three x minus one. The second expression is four x plus three x minus two x plus eight minus one. In both expressions the plus eight and minus one are highlighted blue. The remainder of the expression is highlighted orange. — Image caption,
Algebraic expressions can be simplified by collecting like terms. Positive like terms are often written before the negative like terms.

In algebraic , letters represent unknown numbers (). A variable can have many values or sometimes just one.
An algebraic expression may be by collecting . To reduce the number of in the expression, like terms are added or subtracted.
Like terms may be (number values), a quantity of the same variable or the same combination of variables.
When grouping the like terms together, the order does not matter. However, the positive like terms are often written before the negative like terms.

Video

Watch the video to learn about collecting like terms to simplify expressions.

To play this video you need to enable JavaScript in your browser.

This video can not be played

Recognising and collecting like terms

To recognise , look carefully at each term. Like terms may be:
- (number values)
- terms that use matching
- terms that use matching combinations of variables
Once like terms are identified, they can be collected and grouped together. The is then ready to be simplified.

Examples

Skip image gallery

Image caption,
4𝒙, – 2𝒙 and 3𝒙 are like terms. They are all amounts of 𝒙
Image caption,
Identify the like terms in this expression.
Image caption,
There are two groups of like terms in the expression. 4𝒙, 3𝒙 and – 2𝒙 are like terms as they are all amounts of the same variable, 𝒙. 8 and – 1 are like terms as they are both constants.
Image caption,
The like terms are collected. The sign in front of each term stays with that term. The – 2𝒙 stays as – 2𝒙. The 3𝒙 stays as 3𝒙. The + 8 stays as + 8. The – 1 stays as – 1. It is helpful to write the positive like terms before the negative like terms. This expression with the collected like terms is 4𝒙 + 3𝒙 – 2𝒙 + 8 – 1
Image caption,
Identify the like terms in this expression.
Image caption,
There are two groups of like terms in the expression. – 𝒏 and 3𝒏 are like terms. They are both amounts of the same variable, 𝒏. 5𝒎 and 4𝒎 are like terms. They are both amounts of the variable, 𝒎. 𝒏² is not the same as any other term in the expression. The like terms can be collected.
Image caption,
The like terms are collected. The sign in front of each term stays with that term. The + 3𝒏 stays as + 3𝒏. The – 𝒏 stays as – 𝒏. The + 5𝒎 stays as + 5𝒎. The + 4𝒎 stays as + 4𝒎. It is helpful to write the positive like terms before the negative like terms. This expression written with collected like terms is 𝒏² + 3𝒏 – 𝒏 + 5𝒎 + 4𝒎
Image caption,
Identify the like terms in this expression.
Image caption,
There is one group of like terms in the expression. 7𝒂𝒃, – 𝒂𝒃 and – 4𝒂𝒃 are like terms as they are all amounts of the same combination of variables, 𝒂𝒃. The remaining terms are not the same as any other term in the expression (8𝒂, – 5𝒃 and 3). The like terms can be collected.
Image caption,
The like terms are collected. The sign in front of each term stays with that term. The 7𝒂𝒃 stays as 7𝒂𝒃. The – 𝒂𝒃 stays as – 𝒂𝒃. The – 4𝒂𝒃 stays as – 4𝒂𝒃. This expression written with collected like terms is 7𝒂𝒃 – 𝒂𝒃 – 4𝒂𝒃 + 8𝒂 – 5𝒃 + 3

Question

Identify the like terms in the expression 3𝒙 – 1 + 𝒙 + 10

Simplifying expressions by collecting like terms

To simplify expressions by collecting like terms:

Identify and collect the like terms.
Combine the like terms.
Add or subtract the like terms according to the symbols in the expression.

The simplified expression is equivalent to the original expression. They are .

Examples

Skip image gallery

Image caption,
Simplify the expression.
Image caption,
4𝒙, – 2𝒙 and 3𝒙 are like terms, they are all amounts of 𝒙. They can be collected, writing the positive terms first (4𝒙 and 3𝒙) followed by the negative term (– 2𝒙). The expression can be simplified: 4𝒙 + 3𝒙 – 2𝒙 = 5𝒙
Image caption,
Simplify the expression.
Image caption,
There are two groups of like terms in the expression. 4𝒙, 3𝒙 and – 2𝒙 are like terms and 8 and – 1 are like terms. The like terms are collected and the symbol before each term is moved with each term. The positive like terms are written before the negative like terms. The expression can be simplified by adding the like terms. 4𝒙 + 3𝒙 – 2𝒙 = 5𝒙 and 8 – 1 = 7. The simplified expression is 5𝒙 + 7
Image caption,
Simplify the expression.
Image caption,
There are two groups of like terms in the expression, amounts of 𝒎³ and amounts of 𝒏². 𝒏 is not the same as any other term in the expression. The like terms are collected and the symbol before each term is moved with each term. The expression 5𝒎³ + 4𝒎³ + 3𝒏² – 𝒏² + 𝒏 simplifies to 9𝒎³ + 2𝒏² + 𝒏
Image caption,
Simplify the expression.
Image caption,
There is one group of like terms in the expression, amounts of 𝒂𝒃. The remaining terms are not the same as any other term. The like terms are collected and the symbol before each term is moved with each term. The positive like terms are written before the negative like terms. The expression 7𝒂𝒃 – 𝒂𝒃 – 4𝒂𝒃 + 8𝒂 – 5𝒃 + 3 simplifies to 2𝒂𝒃 + 8𝒂 – 5𝒃 + 3
Image caption,
Simplify the expression.
Image caption,
The two terms are like terms as they are both amounts of the same variable (𝒂³). Variables with no number have a coefficient (a number or symbol multiplied with a variable or an unknown quantity in an algebraic term) of 1, so the term 𝒂³ is one 𝒂³. 5𝒂³ – 1𝒂³ = 4𝒂³

Question

Simplify the expression by collecting the like terms.

The expression seven a minus two b minus a minus five b plus three a plus ten c plus eleven.

Practise collecting like terms

Practise collecting like terms with this quiz. You may need a pen and paper to help you.

Quiz

Real-life maths

A man looking at a shirt. — Image caption,
A buyer who works for a group of clothes shops will add up the sales of different sizes of the same item of clothing to help with buying needs.

Collecting like terms is about totalling the amounts of the same item.

A buyer (a person who chooses what type of goods will be sold by a company) working for a group of clothes shops will add up the sales of different sizes of the same item of clothing, eg a jumper. This is not only to find out total sales but also to help them work out buying needs in the future.

This is necessary because if the buyer just adds up sales based on something simply being 'a jumper’, and not considering its different sizes, this would be unhelpful as not everyone wears the same size. Different sizes are not counted together as they are not 'like sizes'. Businesses need to be able to stock items according to what different customers require.