Algebriac Manipulation
Example 5:
1. Addition and Subtraction of Algebraic Fractions
Steps:
1. Factorise expression where possible
2. Make common denominator
3. Combine into 1 fraction
4. Add or subtract and then simplify where possible
Example 1:
3
4
9
8
9 8 17
2 3 6 6
6
6
Example 2:
3
2
3
4
3
4 3
2
2 2 4 2 2 2 2 2 2 2 2 2
Example 3:
2
3
2 2
3 1
2 2 3 1
1 2
1 2 1 2 1 2
2 4 3 3
5 1
1 2
1 2
1
1
Example 4:
2
3
2
3
23
1
2 2 2 2 2 2
1
2
5
2
5
2
5
4 6 3 2 2 32 2 2 3 2
6
5 2
6 5 10
5 16
3 2 2 3 2 2 3 2 2 3 2 2
Example 6:
2
1
2
1
5 6 2 3 2 3 2 3
2
4
42
2
2 2 3 2 3 2 2 3 2 2 3 2
Steps:
1. Factorise expressions where possible
2. Convert division into multiplication and invert the fraction
!
E.g. "
!
3. If there are any factor that appears in both the numerator and
denominator, cancel them
E.g.
" 4. Multiply the remaining terms: numerator " numerator,
denominator " denominator
"
$
%
$
&
#
$
"
%
$ "
$
#
Example 2:
' # (&' $
'
"
'*
' $ '(&
)'
&('
')
)'
' $ '(&
('
')
(' #
"
)'
'(&
"
'*
'*
Example 3:
,
+
-",
+
+
-
')+
')
'(
%
2
1
1
1
.
/".
/
1 2 1
2 3
2
Common Mistakes
$ )+2
or
$
)+2
Wrong
Example 1:
#
3 2
41
"0
1
2 1
3 2
3
1
"
2 1 3 2
1
2 2
2. Manipulation and division of Algebraic Equations
"
'
&('
(+
'#
"
')
'*
$ )+2
)+2
Correct
15
3. Changing the subject of the formula
Example:
What is changing the subject of the formula?
E.g. Given y + xy = 3x + 2. To make x the subject of the formula, we do
the following:
2− y
y + xy = 3x + 2 xy – 3x = 2 – y x (y – 3) = 2 – y x =
y −3
Make “a” the subject of the formula:
1. remove fraction by multiplying
every term by “bc”:
c
a
× bc = b(a − c) × bc + × bc
b
c
2
2
c = b c(a − c) + ab
In other words, find a formula for x in terms of the others. “x” must not
appear in the formula on the RHS.
2. remove bracket by expansion:
c 2 = b 2 ca − b 2 c 2 + ab
3. move all terms with “a” onto 1
c 2 + b 2 c 2 = b 2 ca + ab
Wrong Method:
side:
c 2 + b 2 c 2 = a (b 2 c + b)
3x + 2 − y
y
This is not the correct way of making x the subject because the RHS still
contains an x term.
4. take out common factor “a” and
c 2 + b 2c 2
=a
b 2c + b
c 2 (1 + b 2 )
∴a =
b(bc + 1)
E.g. y + xy = 3x + 2 xy = 3x + 2 – y x =
Procedure for Changing the subject
Assuming that you want to make “x” the subject of the formula:
Steps:
1. Remove any fractions by multiplying by common multiple of the
denominators or cross multiplying.
2. Remove any brackets by expansion
3. Move all terms containing “x” to 1 side of the equation, all terms
without “x” onto the other side
4. Take out common factor “x” from the side containing x and then
divide as necessary to make x the only term left on that side
3
c
a
= b(a − c ) +
b
c
Steps:
divide: