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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.

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.

- Expanding brackets
- Expand and simplify
- Rearranging equations
- Make x the subject
- Substitution

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...

BUT

^{2}

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

+8x ✔

+8x ✘

-2y ✔

−2y ✘

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

- Highlight the similar terms in the expression and combine them.
- Rewrite the expression.

Simplify

\[8x+5-2x+6\]

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

a)

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

b)

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

2Rewrite the expression.

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

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

Simplify

\[5xy+3y-4-2xy-8y+7\]

Highlight the similar terms in the expression.

a)

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

b)

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

c)

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

Rewrite the expression.

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

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

Simplify

\[6x^{2}y-2x^{2}+4x^{2}y-5x^{2}\]

Highlight the similar terms in the expression.

a) ^{2}y - 2x^{2}^{2}y^{2}

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

b) ^{2}y - 2x^{2} + 4x^{2}y - 5x^{2}

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

Rewrite the expression.

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

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

1. 7 + 2a − 9 + 6a

Show answer

= −2 + 8a

2. 8ab − 8b − 7ab − 3a

Show answer

= ab − 11a

3. −2xy + 3x^{2}y + 7x + 5x^{2}y − 6xy

Show answer

= −8xy + 8x^{2}y + 7x

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

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

Write an expression for the perimeter of the shape.

- 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.

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

Write an expression for the area of the shape.

Read the question carefully and highlight the key information.

**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.

Write an expression and simplify.

\[\frac{(2x+2)(3x+2)}{2}\]

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

\[=3x^{2}+5x+2\]

**Step by step guide:** Expanding Brackets

Sophie is

Emily is three years younger than Sophie

Ameila is four times older than Sophie.

Write an expression for each of their ages.

Read the question carefully and highlight the key information.

We are told that **Sophie is x years old.**

**Emily is three years younger than Sophie**, so three less than

\[x-3\]

**Ameila is four times older than Sophie**, so four lots of

\[4(x-3)\]

We need brackets because we are multiplying all of

Write an expression and simplify.

Sophie is

Emily is

Ameila is

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

Show answer

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

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

Show answer

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

3.

Show answer

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

In order to simplify an algebraic factor you need to:

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

Simplify:

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

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

The HCF of

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.

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

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

Simplify:

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

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

The HCF of ^{2}y^{3}

\[\color{#FF9100}{3x^{2}}\]

Divide the numerator and the denominator by this value.

**Numerator:**

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

**Denominator:**

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

Rewrite the simplified fraction.

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

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

Simplify:

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

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

The HCF of ^{3} - 6xy^{2}y

\[\color{#FF9100}{2x}\]

Divide the numerator and the denominator by this value.

**Numerator:**

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

**Denominator:**

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

Rewrite the simplified fraction.

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

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

Simplify:

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

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

Fully factorise the numerator and the denominator.

**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

Cancel any brackets that are common to the numerator and denominator.

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

Rewrite the simplified fraction.

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

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

1. Simplify:

\[\frac{18ab}{12}\]

Show answer

\[=\frac{3ab}{2}\]

2. Simplify:

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

Show answer

\[=\frac{3b^{2}}{2a}\]

3. Simplify:

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

Show answer

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

4. Simplify:

\[\frac{2x^{2}+6x+4}{4x^{2}-4}\]

Show answer

\[=\frac{x+2}{2x-2}\]

**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.

3x^{2} and 5x^{2} are like terms ✔

2a^{2}b and -5a^{2}b are like terms ✔

BUT

3x^{2} and 5x are not like terms ✘

2a^{2}b 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 ✘

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

Show answer

7f + 3g

(2 marks)

2. Expand and simplify: 4a(a + b) - 2(a^{2} - 2b)

Show answer

2a^{2} + 4ab + 4b

(2 marks)

4. Expand and simplify:

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

Show answer

\[\frac{2x-1}{x-2}\]

(3 marks)

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

- 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.

- Factorising expressions
- Solving equations
- Expanding brackets
- Expand and simplify
- Linear equations
- Quadratic equations
- Simultaneous equations
- Order of operations (BIDMAS)

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