Simplifying Algebraic Expressions

Here is everything you need to know about simplifying algebraic expressions for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to collect like terms, write and simplify expressions, and how to simplify algebraic fractions.

Look out for the algebraic expression worksheets, word problems and exam questions at the end.

What is an algebraic expression?

An algebraic expression is a set of terms that are combined using addition (+), subtraction (−), multiplication (✕) and division (÷).

E.g.

$2x+3y\quad$

or

$2-5y^{2}$

or

$2-5y^{2}+6xy$

Check out our main algebraic expressions lesson for a complete summary of algebraic expressions, and then explore our other lessons for detailed step-by-step guides and worksheets on each type.

Simplify an algebraic expression by collecting the like terms

In order to simplify an algebraic expression we need to ‘collect the like terms’ by grouping together the terms that are similar:

Like terms’ have the same combination of variables and/or numbers as each other, but the coefficients could be different.

For example...

4 and 9 are like terms

3x and 5x are like terms

2ab and -5ab are like terms

BUT

8 and 3x are not like terms

4y and 2x are not like terms

x2 and x are not like terms

When we highlight the like terms, we must include the sign in front of the term.

+8x

+8x

-2y

2y

How to collect the like terms of an algebraic expression

In order to collect the like terms of an algebraic expression:

1. Highlight the similar terms in the expression and combine them.
2. Rewrite the expression.

Simplifying algebraic expressions by collecting the like terms examples

Example 1: with one variable and one constant

Simplify

$8x+5-2x+6$

1. Highlight the similar terms in the expression and combine them.

a) 8x + 5 - 2x + 6

$8x-2x=\color{#00bc89}{3x}$

b) 8x + 5 - 2x + 6

$5+6=\color{#7C4DFF}{11}$

2Rewrite the expression.

$\color{#00bc89}{8x} \color{#7C4DFF}{+5}\color{#00bc89}{-2x}\color{#7C4DFF}{+6}$

$= \color{#00bc89}{3x} \color{#7C4DFF}{+ 11}$

Example 2: with multiple variables and one constant

Simplify

$5xy+3y-4-2xy-8y+7$

a) 5xy + 3y - 4 - 2xy - 8y + 7

$5xy-2xy=\color{#0EE2EF}{3xy}$

b) 5xy + 3y - 4 - 2xy - 8y + 7

$3y - 8y = \color{#00bc89}{-5y}$

c) 5xy + 3y - 4 - 2xy - 8y + 7

$-4+7=\color{#7C4DFF}3$

$\color{#0EE2EF}{5xy}\color{#00bc89}{+3y}\color{#7C4DFF}{-4}\color{#0EE2EF}{-2xy}\color{#00bc89}{-8y}\color{#7C4DFF}{+7}$

$= \color{#0EE2EF}{3xy}\color{#00bc89}{-5y}\color{#7C4DFF}{+3}$

Example 3: with variables to the power of 2

Simplify

$6x^{2}y-2x^{2}+4x^{2}y-5x^{2}$

a) 6x2y - 2x2 + 4x2y - 5x2

$6x^{2}y+4x^{2}y=\color{#0EE2EF}{10x^{2}y}$

b) 6x2y - 2x2 + 4x2y - 5x2

$-2x^{2}-5x^{2}=\color{#FE47EC}{-7x^{2}}$

$\color{#0EE2EF}{6x^{2}y}\color{#FE47EC}{-2x^{2}}\color{#0EE2EF}{+4x^{2}y}\color{#FE47EC}{-5x^{2}}$

$=\color{#0EE2EF}{10x^{2}y}\color{#FE47EC}{-7x^{2}}$

Practice collecting like term questions

1. 7 + 2a − 9 + 6a

= −2 + 8a

2. 8ab − 8b − 7ab − 3a

= ab − 11a

3. −2xy + 3x2y + 7x + 5x2y − 6xy

= −8xy + 8x2y + 7x

Write and simplify algebraic expressions

We can write algebraic expressions to help simplify problems. We will often be able to make a linear equation or a quadratic equation and solve it.

Step by step guide: Solving Equations

How to write and simplify algebraic expressions

1. Read the question carefully and highlight the key information.
2. Write an expression and simplify.

Write and simplify algebraic expressions examples

Example 1: perimeter

Write an expression for the perimeter of the shape.

1. Read the question carefully and highlight the key information.

Key words:

Expression: a set of terms that are combined using (+, −, ✕ and ÷)

Perimeter: the distance around the edge of a shape

We need to add together each of the lengths of the shape.

2Write an expression and simplify.

We then simplify the following expression by adding and subtracting the terms.

\begin{aligned} Perimeter&=\color{#00BC89}{2x}\color{#7C4DFF}{+3}\color{#00BC89}{+x}\color{#7C4DFF}{-2}\color{#00BC89}{+2x}\color{#7C4DFF}{+3}\color{#00BC89}{+x}\color{#7C4DFF}{-2}\\ &=\color{#00BC89}{6x}\color{#7C4DFF}{+2} \end{aligned}

Example 2: area

Write an expression for the area of the shape.

Key words:

Expression: a set of terms that are combined using (+, −, ✕ and ÷)

Area: the 2D space inside a shape.

This shape is a triangle. We know the formula to find the area of a triangle is:

$\text { Area of triangle }=\frac{\text { base } \times \text { height }}{2}$

We need to multiply the base and height of the shape then divide by 2.

$\frac{(2x+2)(3x+2)}{2}$

$= \frac{6x^{2}+10x+4}{2}$

$=3x^{2}+5x+2$

Step by step guide: Expanding Brackets

Example 3: worded problem

Sophie is x years old,

Emily is three years younger than Sophie

Ameila is four times older than Sophie.

Write an expression for each of their ages.

We are told that Sophie is x years old.

Emily is three years younger than Sophie, so three less than x is:

$x-3$

Ameila is four times older than Sophie, so four lots of x − 3 is:

$4(x-3)$

We need brackets because we are multiplying all of x − 3 by 4.

Sophie is x years old.

Emily is x − 3 years old.

Ameila is 4(x − 3) = 4x − 12 years old.

Practice writing and simplifying algebraic expressions questions

1. Write an expression for the perimeter of the shape:

\begin{aligned}\text{ Perimeter } &=2 x+5+x+1+X+3+x+5+x+2+2x+6\\&=8x+22\end{aligned}

2. Write an expression for the area of the shape:

\begin{aligned}&(2x+5)(x+1)=2x^{2}+7x+5\\&\\&(x+5)(x+2)=x^{2}+7x+10\\&\\&\text{ Area }=3x^{2}+14x+15\\&\\&\text {OR}\\&\\&(x+1)(x+3)=x^{2}+4x+3\\&\\&(2x+6)(x+2)=x^{2}+10x+12\\&\\&\text{ Area } = 3x^{2}+14x+15\end{aligned}

3.

Simplify an algebraic fraction

We can write algebraic expressions as fractions and then simplify them.

How to simplify an algebraic fraction

In order to simplify an algebraic factor you need to:

1. Find the highest common factor (HCF) of the numerator and denominator.
2. Divide the numerator and the denominator by this value.
3. Rewrite the simplified fraction.

Simplify algebraic fractions examples

Example 1: variables in the numerator

Simplify:

$\frac{\color{#0EE2EF}{12xy}}{\color{#7C4DFF}{8}}$

1. Find the highest common factor (HCF) of the numerator and denominator.

The HCF of 12xy and 8 is 4

2Divide the numerator and the denominator by this value.

Numerator:

$\color{#0EE2EF}{12xy} \div \color{#FF9100}{4}=\color{#62F030}{3xy}$

Denominator:

$\color{#7C4DFF}8\div \color{#FF9100}{4}=\color{#92009E}2$

3Rewrite the simplified fraction.

$\frac{\color{#0EE2EF}{12xy}}{\color{#7C4DFF}{8}}$

$=\frac{\color{#62F030}{3xy}}{\color{#92009E}2}$

Example 2: variables in the numerator and denominator

Simplify:

$\frac{\color{#FE47EC}{9x^{2}y}}{\color{#159AD3}{15x^{3}}}$

The HCF of 9x2y and 15x3 is:

$\color{#FF9100}{3x^{2}}$

Numerator:

$\color{#FE47EC}{9x^{2}y}\div \color{#FF9100}{3x^{2}}=\color{#62F030}{3y}$

Denominator:

$\color{#159AD3}{15x^{3}}\div \color{#FF9100}{3x^{2}}=\color{#92009E}{5x}$

$\frac{\color{#FE47EC}{9x^{2}y}}{\color{#159AD3}{15x^{3}}}$

$=\frac{\color{#62F030}{3y}}{\color{#92009E}{5x}}$

Example 3: algebraic expression on the numerator

Simplify:

$\frac{\color{#159AD3}{8x^{3}-6xy}}{\color{#FE47EC}{4x^{2}y}}$

The HCF of 8x3 - 6xy and 4x2y is:

$\color{#FF9100}{2x}$

Numerator:

$(\color{#159AD3}{8x^{3}-6xy})\div \color{#FF9100}{2x}=\color{#62F030}{4x^{2}-3y}$

Denominator:

$\color{#FE47EC}{4x^{2}}\div \color{#FF9100}{2x}=\color{#92009E}{2xy}$

$\frac{\color{#159AD3}{8x^{3}-6xy}}{\color{#FE47EC}{4x^{2}y}}$

$=\frac{\color{#62F030}{4x^{2}-3y}}{\color{#92009E}{2xy}}$

Example 4: algebraic expressions on the numerator and the denominator

Simplify:

$\frac{\color{#FE47EC}{x^{2}-2x-15}}{\color{#00bc89}{x^{2}-9}}$

We will need to factorise quadratics to simplify this algebraic fraction.

Numerator:

$\color{#FE47EC}{x^{2}-2x-15}=\color{#62F030}{(x+3)(x-5)}$

Step by step guide: Factorising quadratics

Denominator:

$\color{#00bc89}{x^{2}-9}=\color{#92009E}{(x+3)(x-3)}$

Step by step guide: Difference of two squares

$=\frac{\color{#FF0C3E}{(x+3)}\color{#62F030}{(x-5)}}{\color{#FF0C3E}{(x+3)}\color{#92009E}{(x-3)}}$

$\frac{\color{#FE47EC}{x^{2}-2x-15}}{\color{#00bc89}{x^{2}-9}}$

$=\frac{\color{#62F030}{x-5}}{\color{#92009E}{x-3}}$

Practice simplifying algebraic fractions questions

1. Simplify:

$\frac{18ab}{12}$

$=\frac{3ab}{2}$

2. Simplify:

$\frac{12ab^{3}}{8a^{2}b}$

$=\frac{3b^{2}}{2a}$

3. Simplify:

$\frac{9a^{2}-6ab}{15ab^{2}}$

$=\frac{3a-2b}{5b^{2}}$

4. Simplify:

$\frac{2x^{2}+6x+4}{4x^{2}-4}$

$=\frac{x+2}{2x-2}$

Common Misconceptions

• The sign in front of the term is part of it

When we highlight the like terms, we must include the sign in front of the term.

+8x ✔

8x ✘

-2y ✔

2y ✘

• Terms with a coefficient of 1

For terms with a coefficient of 1 we don’t need to write the 1.

For example:

\begin{aligned} 1x=x\\ 1ab=ab\\ 1y^{2}=y^{2}\\ \end{aligned}

• The commutative property

When adding and multiplying, the order in which we calculate doesn’t matter.

$2x+3x=3x+2x=5x$

And

$2a\times3b=3b\times2a=6ab$

This is not the case for subtracting and dividing.

• Using exponents (powers)

In order for two terms to be ‘like terms’ they need the same combination of variables.

3x2 and 5x2 are like terms

2a2b and -5a2b are like terms ✔

BUT

3x2 and 5x are not like terms

2a2b and -5ab are not like terms ✘

• Using brackets (parentheses)

When multiplying an expression by a value we need to use brackets so that each term is multiplied.

For example:

2(y + 4)

2 ✕ y + 4 ✘

Simplifying algebraic expressions GCSE questions

1. Simplify: 4f - 2e + 3f + 5e

7f + 3g

(2 marks)

2. Expand and simplify: 4a(a + b) - 2(a2 - 2b)

2a2 + 4ab + 4b

(2 marks)

4. Expand and simplify:

$\frac{2x^{2}+7x-4}{x^{2}+2x-8}$

$\frac{2x-1}{x-2}$

(3 marks)

Simplifying algebraic expressions worksheets

Get your free simplifying algebraic expressions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Learning checklist

• Simplify and manipulate algebraic expressions to maintain equivalence by taking out common factors.
• Model situations or procedures by translating them into algebraic expressions.
• Simplify and manipulate algebraic expressions and algebraic fractions.
• Translate simple situations or procedures into algebraic expressions.

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