GCSE Maths

Numbers
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Rounding to the Nearest 10, 100, 1000, and Beyond
Significant Figures
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Percentages
Simple Interest
Rounding and Estimation
Standard Form
Four Operational Rules
Conversion of Units to other Units
Compound Interest
Surds
Rational and Irrational Numbers
Greatest and Least Possible Values
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Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Substituting Values into Formulae
Rearranging Equations
Algebraic Fractions
Expanding Brackets and Simplifying
Expanding Double Brackets
Factorisation
Factorising Quadratic Expressions
Factorising Quadratics with a Coefficient of x Squared Greater than 1
The Quadratic Formula
The Difference of Two Squares
Subject of Formula
Solving Linear Equations
Quadratic Equations
Completing the Square
Sequences and Nth Term
Arithmetic and Geometric Sequences
Quadratic Sequences
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Ratio, Proportion and Rates of Change
7 Topics
Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
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2D Shapes
2D Shapes Quiz
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3D Shapes Quiz
Types of angles
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Angle in Parallel Lines
Triangles
Triangles Quiz
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
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Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
10 Topics
Introduction to Probability
Relative Frequency
Sample Space
Probability of Single Events
Probability of Combined Events
Probability Trees
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Vectors
Mensuration and Calculation
7 Topics
Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Algebraic Notation

GCSE Maths Algebra Algebraic Notation

Algebraic notation is a concise way to express mathematical ideas. In algebra, letters and symbols (a, b, c, d, …, x, y, z, α, β, etc.) represent unknown quantities.

For example, 7 + 7 + 7 + 7 + 7 is the same as 5 \times 7. In algebra, a + a + a + a + a can be written as 5 \times a or simply 5a.

{ b }^{ 2 } + { b }^{ 2 } + { b }^{ 2 } is 3{ b }^{ 2 }, since we have three { b }^{ 2 } added together.

2a stands for a+a and 3b stands for b+b+b.

a^{2} stands for a \times a and a^{3} stands for a \times a \times a.

Infographic titled 'Algebraic Notation'. The graphic is divided into four sections with different color backgrounds:In a green background, 'We group letters together':'a + a + a' is described as meaning '3 lots of a' and equivalent to '3 × a'.'b + b' is described as meaning '2 lots of b' and equivalent to '2 × b'.In an orange background, 'We use indices/powers':The multiplication 'a a' is presented with its result 'a^2', labelled '(a squared)'.The multiplication 'b × b × b' is presented with its result 'b^3', labelled '(b cubed)'.In a yellow background, 'We do not use multiplication signs':'3 × a' is shown as being written as '3a'.'6 × b' as '6b'.Multiplying 'a', 'b', and 'c' together is presented as 'abc', without using multiplication signs.In a grey background, 'We write division using fractional notation':Division 'a ÷ 2' is illustrated as being written as 'a/2' or 'a over 2'.Division 'b ÷ 3' is illustrated as being written as 'b/3' or 'b over 3'.Each section provides examples of how to correctly format and write algebraic expressions.

a \times b is written simply as ab without the multiplication sign in the middle. A \times b \times c = Abc

Algebraic expressions can be simplified by combining like terms. Here are some examples of simplifying algebraic expressions:

  • a + 2a + 3a + 4a + 5a = 15a
  • 7b - b + 3b - 2b + 4b = 11b
  • 3ab + 5ab - 2ab = 6ab
  • 100x^{2} - x^{2} = 99x^{2}
  • a + 2a + 3a + b + 3b + 5b - c + 3c = 6a + 9b + 2c

Coefficients are the numbers that multiply a variable. For example, in the expression 10a + 12b + 3ab - b^{2}:

  • 10 is the coefficient of a
  • 12 is the coefficient of b
  • 3 is the coefficient of ab
  • 1 is the coefficient of b^{2}

Fractions can also be used to represent division. For example, a \div b can be expressed as \frac{a}{b}.

Let’s take a look at some more examples:

Examples

Example 1:

Simplify the following expression:

a + 3a + 4b - 5c + 3b + 5a - 8b + c

First, group the like terms together:

a + 3a + 5a + 4b + 3b - 8b - 5c + c

Now, combine the like terms:

9a - b - 4c

So, the simplified form of the expression is:

9a - b - 4c

Example 2:

Simplify the expression:

4a + 7a - a^{2} + 3ab - a + 5a^{2} - 10ab

Group the like terms together:

The a’s: 4a + 7a - a = 10a

The a^{2}‘s: -a^{2} + 5a^{2} = 4a^{2}

The ab’s: 3ab - 10ab = -7ab

Now, combine the terms to get the simplified expression:

10a + 4a^{2} - 7ab

Note that we can also write the expression in different ways, such as:

4a^{2} + 10a - 7ab

or

4a^{2} - 7ab + 10a

or

-7ab + 10a + 4a^{2}

All of these expressions are correct, as the order of the terms does not affect the overall value of the expression. Keep in mind that, in general, a + b = b + a.

Example 3:

Simplify the following

i) a + 2a + 3a +4a + 5a

ii) 7b - b + 3b - 2b + 4b

iii) 3ab + 5ab - 2ab

iv) 100{ x }^{ 2 } - { x }^{ 2 }

v) a + 2a + 3a + b + 3b + 5b - c + 3c

i) a + 2a + 3a + 4a + 5a = 15a

ii) 7b - b + 3b - 2b + 4b = 11b

iii) 3ab + 5ab - 2ab = 6ab

iv) 100{ x }^{ 2 } - { x }^{ 2 } = 99{ x }^{ 2 }

v) a + 2a + 3a + b + 3b + 5b - c + 3c = 6a + 9b + 2c

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