Formulae 5
Changing the Subject

Sometimes a formula can be rearranged into a more useful format. For example, the formula

F = 1.8C + 32

can be used to convert temperatures in degrees Celsius to degrees Fahrenheit. It can be rearranged into the form

C = . . .

to enable temperatures in degrees Fahrenheit to be converted to degrees Celsius. We say that the formula has been rearranged to make C the subject of the formula.

Worked Examples

1

Rearrange the formula

F = 1.8C + 32

to make C the subject of the formula.

Show me

The aim is to remove all terms from the right hand side of the equation except for the C. First subtract 32 from both sides, which gives

F − 32 = 1.8C

Then dividing both sides by 1.8 gives

= C

So the formula can be rearranged as

C =

2

The distance, s, travelled by a car in time t from initial speed u to final speed v is given by the formula

s =

Make v the subject of the formula.

Show me

First multiply both sides of the formula by 2 to give

2s = (u + v)t

Then divide both sides by t, to give

= u + v

Finally, subtract u from both sides to give

− u = v

So the formula becomes

v = − u

Exercises

Make x the subject of each of the following formulae.

(a)
y = 4x x =
(b)
y = 2x + 3 x =
(c)
y = 4x − 8 x =
(d)
y = x =
(e)
y = x =
(f)
y = x + a x =
(g)
y = x =
(h)
y = ax + c x =
(i)
y = x =
(j)
y = x =
(k)
y = a + b + x x =
(l)
y = x =
(m)
y = abx x =
(n)
y = abx + c x =
(o)
y = x =
(p)
p = x =
(q)
y = (a + x)b x =
(r)
y = x =
(s)
q = x =
(t)
v = x =
(u)
z = a + x =

Ohm's law is used in electrical circuits and states that

V = I R

Write formulae with I and R as their subjects.

I =

R =

Newton's Second law states that F = ma .

Write formulae with m and a as their subjects.

m =

a =

The mean of three numbers x, y and z can be found using the formula

m =

Make z the subject of this formula.

z =

A box with a square base has its volume given by

V = x2 h

and its surface area given by

A = 2x2 + 4xh

(a)

Make h the subject of both formulae.

h =

h =

(b)

Find h if A = 24 cm2 and x = 2 cm.

h = cm
(c)

Find h if V = 250 cm3 and x = 10 cm.

h = cm
(a)

Average speed, v ms−1, time, t seconds and distance, d metres, are related by the formula

d = vt .

Make v the subject of the formula.

v =
(b)

Use the formula to find the average speed for each of the following performances at the Beijing Olympics in 2008. Give your answers correct to 2 decimal places.

(i)

Usain Bolt when he won the men's 100 m race in a time of 9.69 seconds

ms−1
(ii)

Usain Bolt when he won the men's 200 m race in a time of 19.30 seconds.

ms−1
(iii)

The Jamaican relay team for the men's 4 × 100 race in a time of 37.10 s.

ms−1