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?

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?

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