AS: Use of Maths Algebra & Graphs Practice sheet...
5
AS: Use of Maths Algebra & Graphs Practice sheet © Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd Rearranging formulae General points • The aim is to isolate the required subject at one side of the formula – all the other terms should be at the other side. • Always do the same thing to both sides of the equation. • To ‘get rid’ of a term, do the opposite (e.g. to get rid of + mv 2 , subtract mv 2 from both sides). • You may need to remove brackets or multiply through by a number to remove fractions (to simplify the expression) before starting to rearrange the terms. • If the required subject is in a negative term, add this term to both sides to avoid having a negative sign with your subject at the end. • If the subject is in the denominator of a fraction, multiply to ‘bring it to the top’. • If the subject is in a square (or square root), isolate the square (or root), then take the square root (or square) on both sides. • If the subject appears in more than one term, bring these terms together at one side of the equation, then write the subject outside a bracket as a common factor. Examples Method Make t the subject of t d v = d vt = v d t = Multiply both sides by t to ‘bring t to the top’. Divide by v to leave t on its own. Make s the subject of as u v 2 2 2 + = as u v 2 2 2 + = as u v 2 2 2 = − a u v s 2 2 2 − = There is only one term in the formula with s in it (2as). Subtract u 2 from both sides to isolate the term 2as. Divide (the whole of) both sides by 2a to leave s on its own and write with s at the left hand side. Make u the subject of as u v 2 2 2 + = 2 2 2 u as v = − as v u 2 2 − = There is only one term in the formula with u in it (u 2 ) Subtract 2as from both sides to leave only the term involving u on one side of the formula. Take the square root to give the formula for u. Make l the subject of ( ) r l r A + = π 2 r rl A π π + = or r l r A + = π rl r A π π = − 2 l r r A = − π r r A l π π 2 − = r r A l − = π There is more than one method: Remove the bracket or Divide by the term outside Isolate the term with l Write with l at the left hand side.
Transcript of AS: Use of Maths Algebra & Graphs Practice sheet...
AS: Use of Maths Algebra & GraphsPractice sheet
© Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd
Rearranging formulae
General points• The aim is to isolate the required subject at one side of the formula – all the other terms
should be at the other side.• Always do the same thing to both sides of the equation.• To ‘get rid’ of a term, do the opposite (e.g. to get rid of + mv2, subtract mv2 from both sides).• You may need to remove brackets or multiply through by a number to remove fractions
(to simplify the expression) before starting to rearrange the terms.• If the required subject is in a negative term, add this term to both sides to avoid having a
negative sign with your subject at the end.• If the subject is in the denominator of a fraction, multiply to ‘bring it to the top’.• If the subject is in a square (or square root), isolate the square (or root), then take the square
root (or square) on both sides.• If the subject appears in more than one term, bring these terms together at one side of the
equation, then write the subject outside a bracket as a common factor.
Examples Method
Make t the subject of t
dv =
dvt =
v
dt =
Multiply both sides by t to ‘bring t to the top’.
Divide by v to leave t on its own.
Make s the subject of asuv 222 +=
asuv 222 +=
asuv 222 =−
a
uvs
2
22 −=
There is only one term in the formula with s in it (2as).Subtract u2 from both sides to isolate the term 2as.Divide (the whole of) both sides by 2a to leave s on its ownand write with s at the left hand side.
Make u the subject of asuv 222 +=22 2 uasv =−
asvu 22 −=
There is only one term in the formula with u in it (u2)Subtract 2as from both sides to leave only the terminvolving u on one side of the formula.Take the square root to give the formula for u.
Make l the subject of ( )rlrA += π
2rrlA ππ += or rlr
A+=
π
rlrA ππ =− 2 lrr
A=−
π
r
rAl
ππ 2−
= rr
Al −=
π
There is more than one method:
Remove the bracket or Divide by the term outside
Isolate the term with l
Write with l at the left hand side.
AS: Use of Maths Algebra & GraphsPractice sheet
© Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd
Examples MethodMake m the subject of:
mghmvE += 2
2
1
mghmvE 22 2 +=
( )ghvmE 22 2 +=
ghv
Em
2
22 +
=
Multiply every term by 2 to get rid of the fraction.
Write m outside a bracket as a common factor.
Divide by the bracket to leave m on its own.
Some to try
1 Make I the subject of IRV = 2 Make m the subject of cmxy +=
3 Make h the subject of hrV 2π= 4 Make r the subject of hrV 2π=
5 Make u the subject of atuv += 6Make P the subject of
100
PRTI =
7 Make W the subject of ( )WLP += 2 8 Make l the subject of ( )tlL α+= 1
9Make h the subject of
( )2
bahA
+=
10Make a the subject of
( )2
bahA
+=
11 Make m the subject of mumvI −= 12 Make u the subject of mumvI −=
13 Make h the subject of rhrA ππ 22 2 += 14 Make a the subject of 221 atuts +=
15Make d the subject of
24 d
LF
π=
16 Make v the subject of 221 cvE =
17Make T the subject of
µT
c =18 Make C the subject of 325
9 += CF
19 Make r the subject of ( )22 rRA −= π 20Make l the subject of
g
lT π2=
21 Make p the subject of 42222 cmcpE += 22Make R the subject of
21
111
RRR+=
23Make v the subject of
2
2
2
1c
v
mcE
−
=24
Make v the subject of
2
2
1c
v
mvp
−
=
AS: Use of Maths Algebra & GraphsPractice sheet
© Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd
Answers
Other answers are possible – if in doubt, ask your tutor.
1
R
VI =
2
x
cym
−=
32r
Vh
π=
4
h
Vr
π=
5 atvu −= 6
RT
IP
100=
7L
PW −=
2 or
2
2LP − 8
t
Ll
α+=1
9
ba
Ah
+=2
10b
h
Aa −=
2 or
h
hbA −2 11
uv
Im
−=
12
m
Imvu
−=
13
r
rAh
ππ
2
2 2−=
142
)(2
t
utsa
−=
15
F
Ld
π4=
16
c
Ev
2=
17 2cT µ= 18 ( )3295 −= FC
19
ππ AR
r−
=2 20
gT
l2
2
=
π
212
422
c
cmEp
−=
22
21
21
RR
RRR
+=
23 22
1
−=
E
mccv
24
22
2 1
1
cpm
v
+
=
AS: Use of Maths Algebra & GraphsPractice sheet
© Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd
Extra practice
Make x the subject of the following:
1 yx =+13 2 24212 +=− yx 3
2
1+=x
y
4 9243 =++ xy 5 yxy 312 =+ 6 )12(3)3(2 −+=+ yxx
73
2
1=
+
−
y
x 8
1
2
+=x
y9
xy
x=
−3
10y
x5
1
3=
−
113
1=
+x
xy 12
123
+=−x
xy
13 32 += yxz 14
x
bxyz
3=
15 13)1( 2 +=+ abx
16 )32( += xy 17
32
yxzx +=
− 18
xy
1=
19 222 yxz += 20 xyyx 4)2(3)3(2 =+−+ 21 1+= xyz
22
yx
n
y
x21
=+
233
2
3=
−
+
xz
yx 241)12(
4
)1( 22
=++−
yx
AS: Use of Maths Algebra & GraphsPractice sheet
© Nuffield Foundation 2004 AS Use of Maths Scheme of Work Wizard Nelson Thornes Ltd
Answers
1
3
1−=y
x2 yx 25 −= 3 12 −= yx
4 yx 23 −= 5
y
yx
2
13 −=
6 yx 69 −=
7 73 += yx 81
2−=
yx
9
1
3
+=y
x
101
5
3+=
yx
11
3
3
−=y
x12
72
3
−
−=y
yx
13
y
zx
3−=
14
yz
bx
3=
15 1)13( −+= abx
16
2
32 −=y
x17 zyx 32 += 18
2
1
yx =
19 22 yzx −= 20
y
yx
42
3
−=
21 ( )y
zx
21−=
223
)1(
y
ynx
+=
23
7
)(3 yzx
−=
24 1)12(12 2 ++−= yx
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