Rearranging Equations

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.
A=B+5
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:
B=A–5
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Changing the subject (Achieve method)
Make I the subject of the formula
V = IR
Make sure new
subject goes here
V is the subject
of this formula
×R
The formula: V = IR can be written as: I
Going backwards the inverse of this is: I
Notice change of sign
when going backwards
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‚R
V
V
I
=
V or
R
I is the subject of
this formula
E.g. 2: Changing the subject of the formula (Achieve method)
Make n the subject of the formula: m = 2n + 1
Write using individual steps
Make sure new
subject goes here
n
The inverse of this is:
n
×2
+1
÷2
–1
or
Notice that a
fraction is used
for division
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n=
m–1
2
m
m
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Changing the subject of the formula
To make C the subject of the formula:
Write using
individual steps
9C
F=
+ 32
5
C
×9
÷5
+ 32
F
The inverse of this is: C
÷9
x5
– 32
F
OR
Notice need for
brackets and
how a fraction is
used for division
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5(F – 32)
=C
9
E.g. 3: Rearrange the following formula so that a is the subject
V = u + at
a
xt
a
t
+u
-u
V-u
a=
t
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V
V
What number am I thinking of…?
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Your Turn:
(Make x the subject for each question)
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Question
y=x+9
y=x–4
y = 3x
y = 5x + 7
y = 3x – 1
y = 6x + a
y = wx – v
Answer
x=y–9
x=y+4
Your Turn: Rearrange these
1.
2.
3.
4.
5.
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P = 4a + 5
A = be
r
E = u - 4v
d
a=P–5
4
e = Ar
b
u = d(E + 4v)
3)
Make
x
the
subject
p = Q + st
Q = 4cp - st
w x
4c
 4y
t
3) Make x the subject
w  x  4 yt
w x
 4y
w  4 yt  x
t
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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Merit: Rearranging
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
F=
subtract 32:
multiply by 5:
divide by 9:
9C
+ 32
5
9C
F – 32 =
5
5(F – 32) = 9C
5(F – 32)
=C
9
5(F – 32)
C=
9
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+32 is furthest
away from C so
inverse this first
÷5 is now furthest
away from C so
inverse this next
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Formulae involving powers and roots
The length c of the hypotenuse of a right-angled triangle is
given by
c = √a2 + b2
where a and b are the lengths of the shorter sides.
Make a the subject of the formula
square both sides:
subtract b2 from both sides:
square root both sides:
c2 = a2 + b2
c2 – b2 = a2
√c2 – b2 = a
a = √c2 – b2
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Formulae involving powers and roots
The time T needed for a pendulum to make a complete swing
is
T = 2π gl
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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Workings on next page …
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Formulae involving powers and roots
l
g
T = 2π
divide both sides by 2π:
T
=
2π
square both sides:
T2
l
=
g
4π2
multiply both sides by g:
T2g
= l
2
4π
l
g
T2g
l=
4π2
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Make L the subject
Your Turn:
1) Make x the subject
2
 a
2

3
y




x


a
 9 y2
x
a  9 y2 x
x
a
9 y2
2) Make x the subject
2x
5
b
a
3) Make x the subject
2x
2
2

 b5
a

b
 2 xh
a
2 x  a (b  5)
2 xh  a 2  b 2
2
2
a (b  5)
a

b
x
x
2
2h
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4.
D = g2 + c
5.
B=e+ h
g= D–c
h = (B – e)2
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:
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S – 2lh
w =
2l + 2h
E.g. 2) Rearrange to make g the subject:
(r – t) = 6 – 2s
g
 Multiply all by g
g(r – t) = 6 – 2gs Multiply out bracket
gr – gt = 6 – 2gs Collect all g terms
gr – gt + 2gs = 6
g(r – t + 2s) = 6
g=
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6
r – t + 2s
on one side of the
equation and
factorise
Your Turn: Rearrange these:
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1.
ab = 3a + 7
2.
a=e–h
e+5
a=
7
b–3
e = – h – 5a
a–1
3.
s(t – r) = 2(r – 3)
r = st + 6
2+s
4.
e= u–1
d
d=
u
e+1
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.
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
1
1
=
+
R
a
b
aΩ
Make R the subject of the formula
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bΩ
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Formulae involving fractions
1
1
1
=
+
R
a
b
Make R the subject of the formula
multiply through by Rab:
Rab Rab Rab
=
+
R
a
b
simplify:
ab = Rb + Ra
factorize:
ab = R(b + a)
divide both sides by a + b:
ab
=R
a+b
ab
R=
a+b
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Your Turn:
1 1 1
+ =
𝑢 𝑣 𝑓
1: Make v the subject of the formula
𝑢𝑓
V=
𝑢−𝑓
2: Make u the subject of the formula
𝑣𝑓
u=
𝑣−𝑓
3: Make f the subject of the formula
𝑢𝑣
f=
𝑣+𝑢
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