Access Mathematics

Transposition of Formulae

2

Learning objectives

After this session you should be able to: Recall simple formulae triangles to model

simple engineering systems Transpose formulae in which the subject is

contained in more than one term Transpose formulae which contain a root or a

power

3

Recap: Make x the subject

Equation:

3x+2 = 23

3x+2 - 2= 23 -2

3x = 23 - 2

x = (23-2)/3

x=7

Formula:

gx + h = k

gx + h - h = k - h

gx = k- h

x = (k - h)/g

Last lecture we examined the differences between equations and formulae and their subsequent solution protocols:

4

Transposition Of Formulae

The rules are exactly the same as for algebra, except the final result is an algebraic expression instead of a numerical answer.

5

Simple Transposition

In the Science units you will come across very simple formulae for instance Newton’s second law

(mechanics) Electrical charge Ohms Law Density

maF

It

Q

IRV

V

m

6

Recap: Simple Transposition Here the same

rules apply as the letters in the formulae are just numbers in disguise

RIIRV &;

IR

V R

I

V

VI R

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Activity In groups make the subject of the following

formulae the variable in parenthesis for: Density (m)

Electrical Charge (t)

Newton’s second law (a)V

m

It

Q

maF

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Transposition of Elementary formulae

Mathematical SystematicTPRT

I ;100

P

PRT

P

I

100

R

RT

PR

I

100

100

100100 T

PR

I

PR

IT

100

100

T

PR

I

TPR

I

100

Try this one yourselves:

rrc ;2

c

r 22

cr

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Extra terms

Mathematical v=u+at;t v-u=u+at-u v-u=at (v-u)/a=at/a t=(v-u)/a

Systematic v=u+at;t v-u=at (v-u)/a=t

Try this yourself but this time transpose for a instead

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Transposition & substitution

Use any either of the methods to transpose find the the value of R given that:

H=126, t=7 & I=3.RtIH 2

rr

mvT ;

2

2mvTr

Consider:

T

mvr

2

What if m=2, v=5 and T=10

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Example:

The pressure p in a fluid of density at a depth h is given by:

Where pa is the atmos pressure and g is gravitational acceleration.

Make h the subject

hpp a ghpp a g

hpp a g

h

pp a g

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Group Activities Work in groups Discuss the solution for

one the following problems

Select a group member to share your solution with the class

121

; FFF

Pv

ttRR o );1(

bb

bsP ;100

121

;111

RRRR

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Class Discussion/Exercise

(a,b)

TPnRTpV ,;

hh

H

r

R;

mcmxy ;

RrR

VI ;

RhRh

V ;23

2

221

21 ; RRR

RRR

22

222 ;a

a

bae

HcHbhhHa

A ;2

)(

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Subjects with Roots or Powers

In these cases we proceed as before isolating the power or the root first

Thereafter we simply us the inverse operation in order isolate the required variable

i.e. take the root or raise to the power respectively

e.g.

Try

uasuv ;222 22 2 uasv

uasv 22

4

2rA

24r

A

A

r4

15

Transposition inc. Roots/Powers

The same procedure is employed where roots are involved. However to negate the root we raise to the appropriate power:

E.g. LL

T ;2g

g

LT

2 g

LT

2

2

LT

2

2g or 2

2

4T

Lg

Try: hhv ;2g

hv g22 hv

g2

2

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Class Exercise

rrA ;)1 2

cmcE ;)2 2

vmvE ;)3 221

AA

d ;)4

fU

VfE ;

2)5

2

ccab ;)6 22

vmvhmE ;)7 221

max g

aba

k ;12

)822

LRL

k ;412

)922

PQPk ;)10 2221

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Summary Have you met the learning objectives

Specifically are you able to: Recall simple formulae triangles to model simple

engineering systems Transpose formulae in which the subject is contained

in more than one term Transpose formulae which contain a root or a power