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Rearranging Equations

http://www.youtube.com/watch?v=RStSzBUNxBI&feature=related

(rearranging equations)

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Words to learn

Subject of an equation: The letter in front of the equals

sign without any other numbers or letters i.e. ‘y’ is the

subject of y = 3x + 2.

Equation: Two expressions that equal each other.

Inverse: Opposite e.g. the inverse to add is to subtract.

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Using inverse operations

Andy is 5 years older than his brother, Brian. Find a formula

that links their ages.

Using this formula it is easy to find Andy’s age given Brian’s age.

Suppose we want to find Brian’s age given Andy’s age.

Using inverse operations, we can write this formula as:

A = B + 5

B = A – 5

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Changing the subject (Achieve method)

V = IR Make I the subject of the formula

The formula: can be written as: I × R V

V is the subject

of this formula

V = IR

Make sure new

subject goes here

Going backwards the inverse of this is: V ‚ R I or

I is the subject of

this formula

I = V R

Notice change of sign

when going backwards

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Make n the subject of the formula: m = 2n + 1

n + 1 × 2 m

The inverse of this is: m ÷ 2 – 1 n

or

E.g. 2: Changing the subject of the formula (Achieve method)

n = m – 1

2

Make sure new

subject goes here

Write using individual steps

Notice that a

fraction is used

for division

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Changing the subject of the formula

To make C the subject of the formula:

5(F – 32) 9

= C

F = + 32 9C

5

C ÷ 5 × 9 F + 32

The inverse of this is: F x 5 – 32 C ÷ 9

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your notes

Notice need for

brackets and

how a fraction is

used for division

Write using

individual steps

OR

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a

a

V = u + at

V

V

a = V - u

t

x t + u

t - u

E.g. 3: Rearrange the following formula so that a is the subject

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What number am I thinking of…?

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Question Answer

y = x + 9

y = x – 4

y = 3x

y = 5x + 7

y = 3x – 1

y = 6x + a

y = wx – v

Your Turn:

(Make x the subject for each question)

x = y – 9

x = y + 4

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P = 4a + 5 1.

A = be r

2.

E = u - 4v d

3.

4. Q = 4cp - st

a = P – 5

4

e = Ar

b

u = d(E + 4v)

p = Q + st

4c

Your Turn: Rearrange these

3) Make the subject

4

4

4

x

w

w x yt

w yt

x

x

yt

5.

3) Make the subject

4

4

4

x

w

w x yt

w yt

x

x

yt

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Questions to do from the books

Achieve Merit Excellence

Gamma P39 Ex3.04 Q1–21 P39 Ex3.04 Q22–25 P39 Ex3.05

CAT P27 Q193–208 P29 Q209–214

Merit students: do a couple of Merit questions only. We

need to move on to the harder work

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Merit method of Rearranging

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subtract 32: F – 32 = 9C

5

multiply by 5: 5(F – 32) = 9C

divide by 9: 5(F – 32)

9 = C

F = + 32 9C

5

5(F – 32)

9 C =

To rearrange an equation, work from the function furthest away

from the the new variable and do the inverse. (Like getting to the centre of Russian dolls). E.g. make C the subject of

+32 is furthest

away from C so

inverse this first

Merit: Rearranging

÷5 is now furthest

away from C so

inverse this next

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your notes

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Formulae involving powers and roots

The length c of the hypotenuse of a right-angled triangle is

given by

where a and b are the lengths of the shorter sides.

c = √a2 + b2

Make a the subject of the formula

square both sides: c2 = a2 + b2

subtract b2 from both sides: c2 – b2 = a2

a = √c2 – b2

square root both sides: √c2 – b2 = a

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Formulae involving powers and roots

The time T needed for a pendulum to make a complete swing

is

T = 2π l g

where l is the length of the pendulum and g is acceleration

due to gravity.

Make l the subject of the formula

When the variable that we wish to make the subject appears

under a square root sign, we should isolate it on one side of

the equation and then square both sides.

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your notes

Workings on next page …

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Formulae involving powers and roots

T = 2π l g

divide both sides by 2π: T

2π = l

g

square both sides: T2

4π2 = l

g

multiply both sides by g: T2g

4π2 = l

l = T2g

4π2

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your notes

Make L the subject

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2

2

2

2

2

1) Make the subject

3

9

9

9

x

ay

x

ay

x

a y x

ax

y

2) Make x the subject

25

25

2 ( 5)

( 5)

2

xb

a

xb

a

x a b

a bx

2 2

2 2

2 2

2

2

2

a b xh

xh a b

a bx

h

D = g2 + c 4.

B = e + h 5.

g = D – c

h = (B – e)2

Your Turn:

3) Make x the subject

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Equivalent formulae

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Change the subject of the formula 1

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Change the subject of the formula 2

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Questions to do from the books

Achieve Merit Excellence

Gamma P39 Ex3.04 Q1–21 P39 Ex3.04 Q22–25 P39 Ex3.05

CAT P27 Q181 – 192 P27 Q193 – 208 P29 Q209 – 214

Excellence students: The next few slides are more challenging.

Try all, especially the fraction problem as this is mentioned in

the standard.

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Excellence and beyond

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Formulae where the subject appears twice

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Formulae where the subject appears twice

Sometimes the variable that we are making the subject of a

formula appears twice. E.g. 1

S = 2lw + 2lh + 2hw

Make w the subject of the formula.

To do this we must collect all terms containing a w on the

same side of the equals sign.

We can then isolate w by factorizing.

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Formulae where the subject appears twice

S = 2lw + 2lh + 2hw

Swap the left-hand side and the right-hand side so that the

terms with w’s are on the left.

2lw + 2lh + 2hw = S

subtract 2lh from both sides: 2lw + 2hw = S – 2lh

factorize: w(2l + 2h) = S – 2lh

divide by 2l + 2h: w = S – 2lh

2l + 2h

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E.g. 2) Rearrange to make g the subject:

Multiply all by g

Multiply out bracket

Collect all g terms

on one side of the

equation and

factorise

(r – t) = 6 – 2s g

g(r – t) = 6 – 2gs

gr – gt = 6 – 2gs

gr – gt + 2gs = 6

g(r – t + 2s) = 6

g = 6 r – t + 2s

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Your Turn: Rearrange these:

ab = 3a + 7 1.

s(t – r) = 2(r – 3) 3.

e = u – 1 d

4.

a = e – h e + 5

2.

r = st + 6

2 + s

a = 7

b – 3

e = – h – 5a

a – 1

d = u

e + 1

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Formulae involving fractions

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Formulae involving fractions

When a formula involves fractions we usually remove these

by multiplying before changing the subject.

Make R the subject of the formula

For example, if two resistors with a resistance a and b ohms

respectively, are arranged in parallel their total resistance R

ohms can be found using the formula,

1

R =

1

a +

1

b aΩ bΩ

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Formulae involving fractions

multiply through by Rab: = + Rab

R

Rab

a

Rab

b

simplify: ab = Rb + Ra

factorize: ab = R(b + a)

1

R =

1

a +

1

b

divide both sides by a + b: = R ab

a + b

R = ab

a + b

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Make R the subject of the formula

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Your Turn:

1

𝑢+1

𝑣=1

𝑓

1: Make v the subject of the formula

2: Make u the subject of the formula

3: Make f the subject of the formula

V= 𝑢𝑓

𝑢−𝑓

u= 𝑣𝑓

𝑣−𝑓

f= 𝑢𝑣

𝑣+𝑢