Deriving Equations

Deriving Equations

GCSE(F), GCSE(H),

Equations are derived by examining data in a given situation:

• whether the value of one item varies with another;

• 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`

(Hint: instead of changing data items to `x`, use the first letter of the data item as the variable. Constants can be substituted for `m` and `c` as required.)

Examples

1. 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?

Answer: 5.6 kilometers

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

Rewrite as `c=2.85d + 2`.

Substituting into the equation `17.96 = 2.85d + 2`

`17.96 = 2.85d + 2` (next subtract 2 from both sides)

`15.96 = 2.85d` (and divide both sides by 2.85)

`5.6 = d`

2. 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?

Answer: 20 minutes

Create an equation; `text(shift) = text(rate) xx text(parts) + text(warmup)`

`s = rp + w`

`8xx60 = 5xx92 + w` (convert 8 hours to minutes)

`480 = 460 + w`

`w=20`