In this section we discuss the difference between equations, formulae and identities, and then go on to make use of them.

An *equation* contains unknown quantities; for example,

3*x* + 2 = 11

This equation can be solved to determine *x*.

A *formula* links one quantity to one or more other quantities; for example,

*A* = *πr*^{2}

This formula can be used to determine *A* for any given value of *r*.

An *identity* is something that is always true for any values of the variables that are involved; for example,

2(*x* + *y*) ≡ 2*x* + 2*y*

and

(x + y)^2 ≡ x^2 + 2xy + y^2

If any pair of values of *x* and *y* are substituted, then the left hand side of an identity will generate the same value as the right hand side of that identity.

The formula *C* = (*F* – 32) is used to convert temperatures in degrees Fahrenheit to degrees Celsius.

(a)

If *F* = 41, calculate *C*.

*C* = × (41 – 32)

*C* = × 9

*C* = 5

(b)

If *F* = 131, calculate *C*.

*C* = × (131 – 32)

*C* = × 99

*C* = 55

A formula states that *v* = *u* + *at*.

Calculate *v* if *u* = 10, *a* = 6.2 and *t* = 20.

When substituting into equations, you need to be aware that the BODMAS rule applies automatically.

*v* = *u* + *at*

*v* = 10 + 6.2 × 20

*v* = 10 + 124

*v* = 134

Solve the following equations:

(a)

7*x* = 21

7x = | 21 | |

x = | Dividing both sides by 7 | |

x = | 3 |

(b)

*x* – 5 = 12

x – 5 | = 12 | |

x | = 12 + 5 | Adding 5 to both sides |

x | = 17 |

(c)

2*x* + 1 = 6

2x + 1 | = 6 | |

2x | = 6 – 1 | Subtracting 1 from both sides |

2x | = 5 | |

x | = | Dividing both sides by 2 |

x | = 2 |

(d)

5*x* – 8 = 22

5x – 8 | = 22 | |

5x | = 22 + 8 | Adding 8 to both sides |

5x | = 30 | |

x | = | Dividing both sides by 5 |

x | = 6 |

One of the following statements is *not* an identity. Which one?

A | ≡ + | |
---|---|---|

B | x - y | ≡ y - x |

C | x^2 + y^2 | ≡ (x + y)^2 - 2xy |

An identity will be true for any pair of values *x* and *y*. We could test each statement with *x* = 5 and *y* = 10 .

Left-hand-side of A | = | = = = 7.5 |

Right-hand-side of A | = | + = + = 2.5 + 5 = 7.5 |

Therefore LHS of A = RHS of A if *x* = 5 and *y* = 10.

LHS of B | = | x - y = 5 - 10 = -5 |

RHS of B | = | y - x = 10 - 5 = 5 |

Therefore LHS of B ≠ RHS of B if *x* = 5 and *y* = 10.

LHS of C | = x^2 + y^2 = 5^2 + 10^2 = 25 + 100 = 125 |

RHS of C | = (x + y)^2 - 2xy = (5 + 10)^2 - 2 × 5 × 10 = 15^2 - 100 |

= 225 - 100 = 125 |

Therefore LHS of C = RHS of C if *x* = 5 and *y* = 10.

So statement B is *not* an identity. We have *not proved* that A and C are identities, but we know that they are true for certain values of *x* and *y*.