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
(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.)
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`