Equations can be derived :

• when the value of one item varies with another;

• there is a rate (often seen with the word *per*) that multiplies the item;

• a starting value, which is added (or subtracted).

If the change is linear (a straight line), the data items can be laid out as an equation:

result = rate x item + starting value

which corresponds to

`y=mx+c`

A taxi firm charges £2.85 per kilometre, plus a £2 hire charge. If my taxi fare was £17.96, what distance did I travel?

The equation is: `text(cost) = 2.85 xx text(distance) + 2`

Rewrite as | `c` | `=` | `2.85d` | `+` | `2` |

Substituting | `17.96` | `=` | `2.85d` | `+` | `2` |

Subtract 2 from both sides | `15.96` | `=` | `2.85d` | `` | `` |

Divide both sides by 2.85 | `5.6` | `=` | `d` | `` | `` |

Answer: 5.6 kilometers

A company produces a complicated part for a car. The machine takes time to set up before it can produce the parts: after it has been set up, it produces 1 part every 5 minutes.

If the machine produces 92 parts in an 8-hour shift, how long is the set-up time?

Create an equation | shift | = | rate x parts | + | warmup |

replace with letters | `s` | `=` | `rp` | `+` | `w` |

Substitute, using minutes | `8xx60` | `=` | `5xx92` | `+` | `w` |

calculate | `480` | `=` | `460` | `+` | `w` |

Subtract 460 from both sides | `20` | `=` | `` | `` | `w` |

Answer: 20 minutes

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