# Expressions, equations, formulae, functions and identities

## In a nutshell

In maths, and especially in the topic of algebra, the words expression, equation, formula, function and identity is used to describe different mathematical statements. It is important to understand the difference between each of these terms. If you are given a mathematical statement, you should be able to categorise it under one of the terms.

## Expressions

Remember that algebraic terms are a combination of numbers and letters, examples include:

$2x$ |

$-3a$ |

$7n^2$ |

$5xy$ |

Expressions consist of a string of terms, separated by a $+$ or $-$.

##### Examples

$3a+2b+1$ |

$-5x + 2y$ |

$x^2 + 7x + 12$ |

$2xy+ 3y^2 -5x+z$ |

## Equations

Equations consist of two expressions which are equal to each other. If an equation has only one unknown quantity, it can usually be solved to find the value of the unknown variable. If an equation has more than one unknown quantity, then it would have to be solved simultaneously together with another equation. This topic has been covered in 'Solve equations' and 'Simultaneous equations'. Here are some examples of equations:

$2x+3=11$ |

$14=2(x+5)$ |

$x^2+3x=40$ |

$2x+3y=10$ |

## Formulae

A formula is a rule which helps you work something out. It looks like an equation, with an $=$, but will usually have more letters rather than numbers. You would substitute numbers into the formula to work out the value of the variable. You may also need to rearrange the formula, this topic is covered in 'Rearranging formulae'. Here are some examples of formulae.

$F= \frac 9 5 C +32$ | Converts between Fahrenheit and degrees Celsius |

$s=\frac d t$ | Calculates the average speed, given the distance travelled and time taken. |

$v=u+at$ | Calculates the final velocity v of an object, given the initial speed u, acceleration a and time t. |

$\frac 1 R_T = \frac 1 R_1 + \frac 1 R_2$ | Calculates the total resistance $R_T$ of a parallel arrangement of resistors with resistance $R_1$ and $R_2$ |

## Functions

A function is an expression that takes an input value, $x$, performs certain operations, then produces an output value, $y$. A function machine can be represented diagrammatically, as well as with $f(x)$ notation. Here is an example of a function machine.

$x \longrightarrow \boxed {\times 2} \longrightarrow \boxed { +3 } \longrightarrow y$

Use $5$ as an input for the function machine, and it gives $13$ as the output.

$5 \longrightarrow \fcolorbox{black}{white}{$ \times 2 $} \longrightarrow \fcolorbox{black}{white}{ $ +3 $} \longrightarrow 13$

This can be written using $f(x)$ notation as follows:

$f(x) = 2x+3 \\f(5) = 2 \times 5 + 3 = \underline{13}$

## Identities

An identity is an equation which is always true, and works for all values of $x$. Identities have a triple equals sign, $\equiv$.

##### Examples

$0.5a \equiv \frac 1 2 a$ |

$2(x+1) \equiv 2x+2$ |

$x^2-y^2 \equiv (x+y)(x-y)$ |

$2(x-9)+7(x-7) \equiv 9x-67$ |