Deriving Equations

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

$y=mx+c$

Example 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	$17.96$	$=$	$2.85d$	$+$	$2$
Subtract 2 from both sides	$15.96$	$=$	$2.85d$
Divide both sides by 2.85	$5.6$	$=$	$d$

Answer: 5.6 kilometers

Example 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	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