Algebraic Reasoning

An algebraic expression is a set of terms involving constants (numbers) and variables (letters) that are combined using different operators
- The following are all examples of algebraic expressions
  4x²
  2x + y
  3(x - 8)
An equation is a mathematical statement where two expressions are equal to one another
- An equation always contains an equals sign (=)
- It is true for certain values only
- For example, 3x − 1 = 5 is an equation and is only true when x = 2
- Or another example, x² = 9 is an equation and is true only when x = 3 or when x = −3
An identity is a mathematical statement that is true for all values
- For example, 2(3x) ≡ 6x is an identity because it is true for all values of x
  - Note that the symbol for an identity, ≡, is 3 horizontal lines (like an equals sign but with an extra line)

You may come across 'show that' questions that involve some of the following notation

You may be given two different pieces of information about the same thing and asked to show that a particular mathematical statement is true
- For example, you may be told that the volumes of two objects are same and asked to show that one of the lengths is equal to a particular expression
To solve a question of this type
- Find an expression for each item
- Put an equals sign between the two expressions
- Rearrange and simplify to find the statement that you are asked to find

You will often see a statement in words saying that two things are the same as each other.
- Focus on translating this statement into mathematical notation.

The area of the rectangle below is 17 cm².

A rectangle with length (5 + x) cm and width (3 + x) cm

Show that $x^{2} + 8 x = 2$ .

Find an expression for the area of the rectangle by multiplying together the length and the width

$\begin{array}{rcl} (x + 5) (x + 3) & = & x^{2} + 5 x + 3 x + 15 \\ = & x^{2} + 8 x + 15 \end{array}$

You are also told that the area of the rectangle is equal to 17 cm²

Put both of these pieces of information about the area together with an equals sign between them

$x^{2} + 8 x + 15 = 17$

Rearrange by subtracting 15 from both sides to show the equation in the question

^{$x^{2} + 8 x = 2$}

Algebraic Reasoning (Edexcel GCSE Maths: Foundation)