Year 11 General Linear Equations And Their Graphs

Developing A Formula: Setting Up Linear Equations In Two Unknowns

5 practice questions 2 video lessons Theory + worked examples

Theory

A formula connects two or more variables. This page covers building a formula from a real-world context, substituting values, and rearranging to make a different variable the subject — with examples on cinema tickets, taxi fares, manufacturing costs, and coin tins.

A formula is a rule that connects two or more variables. Once you have the formula, you can substitute any values to compute the answer, or rearrange the formula to make a different variable the subject.

Most word problems give you one or more rates (e.g.\ $$ 5$ per coffee, $$ 2$ per kilometre), sometimes a fixed amount (e.g.\ $$ 30$ per month, $$ 5$ flagfall), and a total you want to express. The general pattern is to multiply each rate by the matching quantity and add up everything, plus any fixed amount.

To rearrange a formula and make a different variable the subject, treat it like an equation and use the same balance moves (add, subtract, multiply, divide both sides). The goal is to isolate the new subject.

The general pattern: total = sum of (rate × quantity), plus any fixed amount.

Rearranging

C = 5 x + 2 y

to make

y

the subject, step by step.

The general formula pattern for cost or value problems:

total = ({rate}_{1} \cdot {quantity}_{1}) + ({rate}_{2} \cdot {quantity}_{2}) + fixed

total = r_{1} q_{1} + r_{2} q_{2} + fixed

Common geometric formulas built from the same idea:

Rectangle perimeter:

P = 2 L + 2 W

P = 2 L + 2 W

Triangle area:

A = \frac{1}{2} b h

A = \frac{1}{2} b h

Trapezium area:

A = \frac{1}{2} (a + b) h

A = \frac{1}{2} (a + b) h

Rearranging example. From $C = 5 x + 2 y$ , to make $y$ the subject: subtract $5 x$ from both sides to get $C - 5 x = 2 y$ , then divide both sides by $2$ to get $y = \frac{C - 5 x}{2}$ .

How to build and use a formula

Identify the variables. Choose a letter for each unknown quantity and one for the total.
Write the formula by adding up rate × quantity for each item, plus any fixed amount.
Use the formula by substituting given numbers.
Rearrange if needed to find a different variable — use the balance moves to isolate the new subject.

Unit check. Cents and dollars must be in the same unit. $50$ c $= $ 0.50$ . For a coin tin, $V = 0.50 x + 2 y$ , not $50 x + 2 y$ . Always check unit consistency before substituting.

Example 1 — Two-Item Formula

A cinema charges

$ 18

per adult ticket and

$ 12

per child ticket. Let

a

be the number of adults and

c

the number of children. Write a formula for total cost

C

. Find

C

for

4

adults and

6

children.

Solution

Formula: $C = 18 a + 12 c$ . Substitute $a = 4$ , $c = 6$ .

$C$	$=$	$18 (4) + 12 (6)$
$C$	$=$	$72 + 72$
$C$	$=$	$$ 144$

C = 144

Example 2 — Fixed Fee Plus Rate

A taxi charges a

$ 5

flagfall plus

$ 2.50

per km. Write a formula for cost

C

for a trip of

d

km, then find

C

for an

8

km trip.

Solution

Formula: $C = 5 + 2.50 d$ . Substitute $d = 8$ .

$C$	$=$	$5 + 2.50 (8)$
$C$	$=$	$5 + 20$
$C$	$=$	$$ 25$

C = 25

Example 3 — Rearrange a Formula

The cost of producing

x

shirts and

y

caps is

C = 5 x + 2 y

. Rearrange to make

y

the subject. Then find

y

when

C = 50

and

x = 4

Solution

Subtract $5 x$ from both sides, then divide by $2$ .

$C$	$=$	$5 x + 2 y$
$C - 5 x$	$=$	$2 y$
$y$	$=$	$\frac{C - 5 x}{2}$

With $C = 50$ , $x = 4$ :

$y$	$=$	$\frac{50 - 20}{2}$
$y$	$=$	$15$

y = 15

Example 4 — Coin Formula

A money tin holds

50

c coins and

$ 2

coins. Let

x

be the number of

50

c coins and

y

the number of

$ 2

coins. Write a formula for value

V

in dollars, then find

V

for

12

50

c and

8

$ 2

Solution

Use dollars throughout: $50$ c $= $ 0.50$ . Formula: $V = 0.50 x + 2 y$ .

$V$	$=$	$0.50 (12) + 2 (8)$
$V$	$=$	$6 + 16$
$V$	$=$	$$ 22$

V = 22

Common pitfalls

Pairing the wrong rate with the wrong quantity. If lamingtons cost

$ 2.50

and tarts cost

$ 1.75

, then

C = 2.50 x + 1.75 y

where

x

is the number of lamingtons. Don't pair lamingtons with the tart price.

Mixing dollars and cents.

50

c is

$ 0.50

. If the total is in dollars, every term must be in dollars. The formula

V = 50 x + 2 y

treats

50

c coins as

$ 50

each — wildly wrong.

Forgetting the fixed term when rearranging. When changing the subject, every term moves.

C = a + b x

rearranged for

x

gives

x = \frac{C - a}{b}

— the

a

must come across before you divide.

Wrong sign when moving across. When

5 x

crosses the equals sign, it becomes

- 5 x

. Watch the sign — a flipped sign turns the formula into the wrong rearrangement.

Frequently asked questions

What is a formula?

A formula is a rule that connects two or more variables. Once you have the formula, you can substitute any values to compute the answer, or rearrange the formula to make a different variable the subject.

How do I build a formula from a word description?

Identify the rates (cost per item, dollars per kilometre), any fixed amounts (flagfall, monthly fee), and what total you want to find. The general pattern is: total equals rate times quantity, plus any fixed amount. Add a similar term for each item.

How do I rearrange a formula?

Treat the formula like an equation and use the same balance moves: add, subtract, multiply, or divide both sides by the same value. The goal is to isolate the variable you want to make the subject.

What is the difference between 50c and dollar 0.50 in a formula?

They are the same amount, just in different units. The trap is mixing units in one formula. If the total value V is in dollars, write 0.50x (where x is the number of 50c coins), not 50x. Always check that every term in a formula uses the same unit.

What does 'make y the subject' mean?

It means rearrange the equation so that y is alone on one side, equal to an expression involving everything else. For C equals 5x plus 2y, making y the subject gives y equals open bracket C minus 5x close bracket over 2.

What is the formula for a rectangle's perimeter?

Perimeter equals 2 times length plus 2 times width, or P equals 2L plus 2W. This adds up the four sides: two lengths and two widths.