Formulae 3
Substitution into Formulae

The process of replacing the letters in a formula is known as substitution.

Worked Examples

1

The length of a metal rod is l cm. The length changes with temperature and can be found by the formula

l = 40 + 0.02T

where T is the temperature.

Find the length of the rod when

(a)

T = 50 °C

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Using T = 50 gives

l= 40 + 50 × 0.02
= 40 + 1
= 41 cm
(b)

T = –10 °C

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Using T = −10 gives

l= 40 + (−10) × 0.02
= 40 + (−0.2)
= 40 − 0.2
= 39.8 cm
2

The profit in £ made by a salesman when he sells n books is calculated by the formula

P = 4n − 50

Find the profit if he makes

(a)

30 sales

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Here n = 30 so the formula gives

P= 4 × 30 − 50
= 120 − 50
= 70

So the salesman's profit after 30 sales is £70.
(b)

9 sales

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Here n = 9 so the formula gives

P= 4 × 9 − 50
= 36 − 50
= −14

He makes a loss of £14 if only 9 sales are made.
3

If x * y is defined by

x * y = x2 − 4y2

what is the value of x * y when

(a)

x = 4 ,   y = −2

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4 * (−2)= 42 − 4(−2)2
= 16 − 4 × 4
= 16 − 16
= 0
(b)

x = −2 ,   y = 4 ?

Show me
(−2) * 4= (−2)2 − 4 × 42
= 4 − 4 × 16
= 4 − 64
= −60

Exercises

The formula below is used to convert temperatures in degrees Celsius to degrees Fahrenheit, where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius.

F = 1.8 C + 32

Find F if:

(a)
C = 10 F =
(b)
C = 20 F =
(c)
C = −10 F =
(d)
C = −5 F =
(e)
C = −20 F =
(f)
C = 15 F =

The formula

s = (u + v)t

is used to calculate the distance, s, that an object travels if it starts with a velocity u and has a velocity v, t seconds later.

Find s if:

(a)

u = 2,   v = 8,   t = 2

s =
(b)

u = 3,   v = 5,   t = 10

s =
(c)

u = 1.2,   v = 3.8,   t = 4.5

s =
(d)

u = −4,   v = 8,   t = 2

s =
(e)

u = 4,   v = −8,   t = 5

s =
(f)

u = 1.6,   v = 2.8,   t = 3.2

s =

The formula

P = 120n − 400

gives the profit, P in £ , made when n phones are sold in a day at a shop.

Find P if:

(a)
n = 1 P =
(b)
n = 3 P =
(c)
n = 4 P =
(d)
n = 10 P =

How many phones must be sold to make a profit?

(e)
phones

Work out the value of each function by substituting the values given, without using a calculator.

Give your answer in the form e.g.: 5, -5 if there are two solutions, entering positive numbers first.

(a)

V = p2 + q2 p = 8 and q = 4

V =
(b)

p = a2 − b2 a = 10 and b = 7

p =
(c)

z = x = 10 and y = 6

z =
(d)

Q = x = 15 and y = 6

Q =
(e)

P = x = 4 and y = −10

P =
(f)

Q = a = 100 and b = 4

Q =
(g)

V = x = 2, y = −5 and z = 8

V =
(h)

R = + a = 4 and b = 2

R =
(i)

S = + a = 3, b = 4 and c = 16

S =
(j)

R = 0.2a + 0.4b a = 10 and b = 20

R =
(k)

T = + a = −20 and b = 40

T =
(l)

C = a = 10 and b = −5

C =
(m)

P = 2 x = 10 and y = 4

P =
(n)

A = a = 2, b = 3 and c = 100

A =
(o)

X = a = 10, b = 1.7 and c = 2.1

X =
(p)

z = x = −3 and y = 4

z =
(q)

P = a = −10 and b = 6

P =
(r)

Q = x = −10, y = 5 and z = 10

Q =

The formula to convert temperatures from degrees Fahrenheit (°F) into degrees Celsius (°C) is

C = (F − 32)

Calculate the temperature in degrees Celsius which is equivalent to a temperature of −4 °F.

°C

Given that m = , p = , t = −2, calculate

(a)
mp + t
(b)

Challenge!

There are 10 bank notes altogether. They consist of £10, £20 and £50 notes. If the total value of the notes is £180, find the number of each type of notes.