MTH1W Equations in Context and Rearranging Formulas: Notes and Practice

You're checking the weather for a trip to Florida and the forecast says 86 degrees. That is Fahrenheit, and the formula linking the two scales is $F = 1.8 C + 32$ . You could substitute $F = 86$ and solve an equation for $C$ , and do it again for tomorrow's forecast, and again the day after. Or you could rearrange the formula once, so that it starts with $C =$ , and then just plug in each forecast. Both skills in this lesson come from the same place: the inverse operations you used to solve equations. First you'll turn situations into equations; then you'll use the same moves on formulas full of letters.

Creating an equation from a situation

Define a variable: say exactly what it stands for, with units ("let $d$ be the distance driven, in kilometres").
Write an equation: translate the facts, making sure both sides measure the same thing (dollars = dollars, metres = metres).
Solve with inverse operations.
Check that the answer works in the original situation and makes sense (a number of people can't be 2.5; a length can't be negative).
Answer in a sentence, with units.

Here is the method on a real decision. A moving truck rents for 40 dollars a day plus 0.35 dollars per kilometre, and you have budgeted 75 dollars for one day. Let $d$ be the distance driven in kilometres. The kilometre charge is $0.35 d$ and the daily fee is 40, so the total cost is $40 + 0.35 d$ , and that must equal the budget: $40 + 0.35 d = 75$ . Subtract 40 from both sides: $0.35 d = 35$ . Divide by 0.35: $d = 100$ . Check: $40 + 0.35 (100) = 40 + 35 = 75$ . You can drive 100 km and stay on budget.

Fig 1The 75 dollar budget drawn to scale: 40 dollars for the daily fee and 35 dollars for kilometres, which splits into 10 blocks of 10 km at 3.50 dollars each, so d = 100 km.

Situation	Variable	Equation
A taxi charges a 4 dollar base fare plus 2 dollars per km. The ride cost 22 dollars.	k = distance in km	$4 + 2 k = 22$
Three consecutive integers add to 84.	n = smallest integer	$n + (n + 1) + (n + 2) = 84$
A rectangle's length is 3 m more than its width; its perimeter is 26 m.	w = width in m	$2 w + 2 (w + 3) = 26$
You have 150 dollars saved and save 20 dollars a week. When will you have 390 dollars?	t = time in weeks	$150 + 20 t = 390$

Each situation, a clearly defined variable, and the equation that models it.

Fig 2The rectangle from the table, drawn to scale: width w, length w + 3, and the perimeter equation 2w + 2(w + 3) = 26 solved to give a garden 5 m wide and 8 m long.

Check the situation, not just the equation

An answer can satisfy the equation and still not answer the question. If you solve for the number of tickets and get

12.4

, you can't buy 12.4 tickets: decide whether the question needs 12 (to stay under a budget) or 13 (to reach a goal). Also check that the units match: a rate in dollars per hour needs a time in hours, not minutes.

Quick checkTry it before you read on

A gym charges 30 dollars to join plus 12 dollars per class, and you have 90 dollars to spend. With c as the number of classes, which equation fits?

Quick checkTry it before you read on

Solving for the number of 8-dollar tickets you can buy with 100 dollars gives

t = 12.5

. How many tickets can you actually buy?

tickets

VideoKhan Academy reads a word problem, chooses a variable, writes an equation that models the situation and then solves it, the same steps as this section.Watch on YouTube

Rearranging formulas

A formula is an equation that links two or more variables, like $P = 2 l + 2 w$ for the perimeter of a rectangle. The variable by itself on one side is the subject of the formula; here it is $P$ . To make a different variable the subject, treat every other letter as if it were a number and use exactly the inverse operations you'd use to solve an equation. Ask: what does the formula do to $w$ ? It multiplies $w$ by 2 and then adds $2 l$ . To get $w$ alone, undo those in reverse order: subtract $2 l$ from both sides, then divide both sides by 2.

Fig 3Rearranging P = 2l + 2w for w: the formula multiplies w by 2 then adds 2l, so undo by subtracting 2l then dividing by 2, and check with P = 30 m and l = 9 m.

F F - 32 \frac{F - 32}{1.8} = 1.8 C + 32 = 1.8 C = C subtract 32 divide by 1.8

Making C the subject of the temperature formula.

Formula	Solved for	Rearranged	Inverse steps
$d = s t$ (distance, speed, time)	$t$	$t = \frac{d}{s}$	divide by $s$
$A = lw$ (rectangle area)	$w$	$w = \frac{A}{l}$	divide by $l$
$C = 2 π r$ (circumference)	$r$	$r = \frac{C}{2 π}$	divide by $2 π$
$P = 2 l + 2 w$	$w$	$w = \frac{P - 2 l}{2}$	subtract $2 l$ , divide by 2
$F = 1.8 C + 32$	$C$	$C = \frac{F - 32}{1.8}$	subtract 32, divide by 1.8
$y = m x + b$	$b$	$b = y - m x$	subtract $m x$

Dividing by a variable

When you divide by a letter, the rearranged formula only works when that letter is not zero.

t = \frac{d}{s}

makes no sense for a speed of

s = 0

: if you aren't moving, no amount of time covers the distance. In real formulas the variable you divide by is usually a length, speed or rate that is positive, but say so when it matters.

ASubstitute first, then solve

Put the known numbers in, then solve the equation that is left.
$150 = 60 t$ , so $t = 150 \div 60 = 2.5$ hours.
Best when you need one answer.

BRearrange first, then substitute

Make the unknown the subject: $t = \frac{d}{s}$ .
Then $t = \frac{150}{60} = 2.5$ hours, and the same formula handles every other trip.
Best when you need many answers, or a formula to use in a spreadsheet or code.

Quick checkTry it before you read on

Make

m

the subject of

y = m x + b

Quick checkTry it before you read on

Rearrange

P = a + b + c

to make

c

the subject. Write your answer starting with c =

Equations in Context and Rearranging Formulas

Key things to remember

Creating an equation from a situation

Rearranging formulas

Worked examples

Where marks get lost

Flashcards 8

Practice