Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > Algebraic Expressions and Identities - Exercise 6.2
Algebraic Expressions and Identities - Exercise 6.2 | Mathematics (Maths) Class 8 PDF Download
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Page 1
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Q u e s t i o n : 3
Add the following algebraic expressions:
i 3a
2
b, - 4a
2
b, 9a
2
b
ii
2
3
a,
3
5
a, -
6
5
a
iii 4xy
2
- 7x
2
y, 12x
2
y - 6xy
2
, - 3x
2
y +5xy
2
iv
3
2
a -
5
4
b +
2
5
c,
2
3
a -
7
2
b +
7
2
c,
5
3
a +
5
2
b -
5
4
c
v
11
2
xy +
12
5
y +
13
7
x, -
11
2
y -
12
5
x -
13
7
xy
vi
7
2
x
3
-
1
2
x
2
+
5
3
,
3
2
x
3
+
7
4
x
2
-x +
1
3
,
3
2
x
2
-
5
2
x -2
S o l u t i o n :
i To add the like terms, we proceed as follows:
3a
2
b + -4a
2
b +9a
2
b = 3a
2
b -4a
2
b +9a
2
b = (3 -4 +9)a
2
b Distributive law = 8a
2
b
ii To add the like terms, we proceed as follows:
2
3
a +
3
5
a + -
6
5
a =
2
3
a +
3
5
a -
6
5
a =
2
3
+
3
5
-
6
5
a Distributive law =
1
15
a
( ) ( )
( ) ( ) ( )
Page 2
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Q u e s t i o n : 3
Add the following algebraic expressions:
i 3a
2
b, - 4a
2
b, 9a
2
b
ii
2
3
a,
3
5
a, -
6
5
a
iii 4xy
2
- 7x
2
y, 12x
2
y - 6xy
2
, - 3x
2
y +5xy
2
iv
3
2
a -
5
4
b +
2
5
c,
2
3
a -
7
2
b +
7
2
c,
5
3
a +
5
2
b -
5
4
c
v
11
2
xy +
12
5
y +
13
7
x, -
11
2
y -
12
5
x -
13
7
xy
vi
7
2
x
3
-
1
2
x
2
+
5
3
,
3
2
x
3
+
7
4
x
2
-x +
1
3
,
3
2
x
2
-
5
2
x -2
S o l u t i o n :
i To add the like terms, we proceed as follows:
3a
2
b + -4a
2
b +9a
2
b = 3a
2
b -4a
2
b +9a
2
b = (3 -4 +9)a
2
b Distributive law = 8a
2
b
ii To add the like terms, we proceed as follows:
2
3
a +
3
5
a + -
6
5
a =
2
3
a +
3
5
a -
6
5
a =
2
3
+
3
5
-
6
5
a Distributive law =
1
15
a
( ) ( )
( ) ( ) ( )
iii To add, we proceed as follows:
4xy
2
-7x
2
y + 12x
2
y + -6xy
2
+ -3x
2
y +5xy
2
= 4xy
2
-7x
2
y +12x
2
y -6xy
2
-3x
2
y +5xy
2
= 4xy
2
-6xy
2
+5xy
2
-7x
2
y +12x
2
y -3x
2
y Collecting like terms = 3xy
2
+2x
2
y
iv To add, we proceed as follows:
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a +
2
3
a +
5
3
a -
5
4
b -
7
2
b +
5
2
b +
2
5
c +
7
2
c -
5
4
c Collecting like terms =
23
6
a -
9
4
b +
53
20
c
v To add, we proceed as follows:
11
2
xy +
12
5
y +
13
7
x + -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy +
12
5
y +
13
7
x -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy -
13
7
xy +
12
5
y -
11
2
y +
13
7
x -
12
5
x Collecting like terms) =
51
14
xy -
31
10
y -
19
35
x Combining like terms)
vi To add, we proceed as follows:
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
+
3
2
x
3
-
1
2
x
2
+
7
4
x
2
+
3
2
x
2
-x -
5
2
x +
5
3
+
1
3
-2 Collecting like terms = 5x
3
+
11
4
x
2
-
7
2
x
Q u e s t i o n : 4
Subtract:
i - 5xy from 12xy
ii 2a
2
from - 7a
2
iii 2a - b from 3a - 5b
iv 2x
3
- 4x
2
+ 3x + 5 from 4x
3
+ x
2
+ x + 6
v
2
3
y
3
-
2
7
y
2
-5 from
1
3
y
3
+
5
7
y
2
+y -2
vi
3
2
x -
5
4
y -
7
2
z from
2
3
x +
3
2
y -
4
3
z
vii x
2
y -
4
5
xy
2
+
4
3
xy from
2
3
x
2
y +
3
2
xy
2
-
1
3
xy
viii
ab
7
-
35
3
bc +
6
5
ac from
3
5
bc -
4
5
ac
S o l u t i o n :
(i) 12xy -(-5xy) = 12xy +5xy = 17xy
ii -7a
2
- 2a
2
= -7a
2
-2a
2
= -9a
2
(iii) (3a -5b)-(2a -b) = (3a -5b)-2a +b = 3a -5b -2a +b = 3a -2a -5b +b = a -4b
iv 4x
3
+x
2
+x +6 - 2x
3
-4x
2
+3x +5 = 4x
3
+x
2
+x +6 -2x
3
+4x
2
-3x -5 = 4x
3
-2x
3
+x
2
+4x
2
+x -3x +6 -5 Collecting like terms = 2x
3
+5x
2
-2x +1 Combining like
v
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
-
2
7
y
2
-5 =
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
+
2
7
y
2
+5 =
1
3
y
3
-
2
3
y
3
+
5
7
y
2
+
2
7
y
2
+y -2 +5 Collecting like terms = -
1
3
y
3
+y
2
+y +3 Combining like terms
vi
2
3
x +
3
2
y -
4
3
z -
3
2
x -
5
4
y -
7
2
z =
2
3
x +
3
2
y -
4
3
z -
3
2
x +
5
4
y +
7
2
z =
2
3
x -
3
2
x +
3
2
y +
5
4
y -
4
3
z +
7
2
z Collecting like terms = -
5
6
x +
11
4
y +
13
6
z Combining like terms
vii
2
3
x
2
y +
3
2
xy
2
-
1
3
xy - x
2
y -
4
5
xy
2
+
4
3
xy =
2
3
x
2
y +
3
2
xy
2
-
1
3
xy -x
2
y +
4
5
xy
2
-
4
3
xy =
2
3
x
2
y -x
2
y +
3
2
xy
2
+
4
5
xy
2
-
1
3
xy -
4
3
xy Collecting like terms) = -
1
3
x
2
y +
23
10
xy
2
-
5
3
xy Combining like terms
viii
3
5
bc -
4
5
ac -
ab
7
-
35
3
bc +
6
5
ac =
3
5
bc -
4
5
ac -
ab
7
+
35
3
bc -
6
5
ac =
3
5
bc +
35
3
bc -
4
5
ac -
6
5
ac -
ab
7
Collecting like terms) =
184
15
bc -2ac -
ab
7
Combining like terms
Q u e s t i o n : 5
Take away:
i
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x from
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
ii
5a
2
2
+
3a
3
2
+
a
3
-
6
5
from
1
3
a
3
-
3
4
a
2
-
5
2
iii
7
4
x
3
+
3
5
x
2
+
1
2
x +
9
2
from
7
2
-
x
3
-
x
2
5
iv
y
3
3
+
7
3
y
2
+
1
2
y +
1
2
from
1
3
-
5
3
y
2
v
2
3
ac -
5
7
ab +
2
3
bc from
3
2
ab -
7
4
ac -
5
6
bc
S o l u t i o n :
i The difference is given by:
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x =
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
+
4
5
x
3
-
5
6
-
3
2
x
=
x
3
3
+
4
5
x
3
-
5
2
x
2
-
6
5
x
2
+
3
5
x -
3
2
x +
1
4
-
5
6
Collectingliketerms
=
5+12
15
x
3
+
-25-12
10
x
2
+
6-15
10
x +
6-20
24
=
17
15
x
3
-
37
10
x
2
-
9
10
x -
7
12
Combiningliketerms
ii The difference is given by:
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
+
3a
3
2
+
a
3
-
6
5
=
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
-
3a
3
2
-
a
3
+
6
5
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( (
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( (
( ) ( ) ( ) ( ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
Page 3
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Q u e s t i o n : 3
Add the following algebraic expressions:
i 3a
2
b, - 4a
2
b, 9a
2
b
ii
2
3
a,
3
5
a, -
6
5
a
iii 4xy
2
- 7x
2
y, 12x
2
y - 6xy
2
, - 3x
2
y +5xy
2
iv
3
2
a -
5
4
b +
2
5
c,
2
3
a -
7
2
b +
7
2
c,
5
3
a +
5
2
b -
5
4
c
v
11
2
xy +
12
5
y +
13
7
x, -
11
2
y -
12
5
x -
13
7
xy
vi
7
2
x
3
-
1
2
x
2
+
5
3
,
3
2
x
3
+
7
4
x
2
-x +
1
3
,
3
2
x
2
-
5
2
x -2
S o l u t i o n :
i To add the like terms, we proceed as follows:
3a
2
b + -4a
2
b +9a
2
b = 3a
2
b -4a
2
b +9a
2
b = (3 -4 +9)a
2
b Distributive law = 8a
2
b
ii To add the like terms, we proceed as follows:
2
3
a +
3
5
a + -
6
5
a =
2
3
a +
3
5
a -
6
5
a =
2
3
+
3
5
-
6
5
a Distributive law =
1
15
a
( ) ( )
( ) ( ) ( )
iii To add, we proceed as follows:
4xy
2
-7x
2
y + 12x
2
y + -6xy
2
+ -3x
2
y +5xy
2
= 4xy
2
-7x
2
y +12x
2
y -6xy
2
-3x
2
y +5xy
2
= 4xy
2
-6xy
2
+5xy
2
-7x
2
y +12x
2
y -3x
2
y Collecting like terms = 3xy
2
+2x
2
y
iv To add, we proceed as follows:
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a +
2
3
a +
5
3
a -
5
4
b -
7
2
b +
5
2
b +
2
5
c +
7
2
c -
5
4
c Collecting like terms =
23
6
a -
9
4
b +
53
20
c
v To add, we proceed as follows:
11
2
xy +
12
5
y +
13
7
x + -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy +
12
5
y +
13
7
x -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy -
13
7
xy +
12
5
y -
11
2
y +
13
7
x -
12
5
x Collecting like terms) =
51
14
xy -
31
10
y -
19
35
x Combining like terms)
vi To add, we proceed as follows:
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
+
3
2
x
3
-
1
2
x
2
+
7
4
x
2
+
3
2
x
2
-x -
5
2
x +
5
3
+
1
3
-2 Collecting like terms = 5x
3
+
11
4
x
2
-
7
2
x
Q u e s t i o n : 4
Subtract:
i - 5xy from 12xy
ii 2a
2
from - 7a
2
iii 2a - b from 3a - 5b
iv 2x
3
- 4x
2
+ 3x + 5 from 4x
3
+ x
2
+ x + 6
v
2
3
y
3
-
2
7
y
2
-5 from
1
3
y
3
+
5
7
y
2
+y -2
vi
3
2
x -
5
4
y -
7
2
z from
2
3
x +
3
2
y -
4
3
z
vii x
2
y -
4
5
xy
2
+
4
3
xy from
2
3
x
2
y +
3
2
xy
2
-
1
3
xy
viii
ab
7
-
35
3
bc +
6
5
ac from
3
5
bc -
4
5
ac
S o l u t i o n :
(i) 12xy -(-5xy) = 12xy +5xy = 17xy
ii -7a
2
- 2a
2
= -7a
2
-2a
2
= -9a
2
(iii) (3a -5b)-(2a -b) = (3a -5b)-2a +b = 3a -5b -2a +b = 3a -2a -5b +b = a -4b
iv 4x
3
+x
2
+x +6 - 2x
3
-4x
2
+3x +5 = 4x
3
+x
2
+x +6 -2x
3
+4x
2
-3x -5 = 4x
3
-2x
3
+x
2
+4x
2
+x -3x +6 -5 Collecting like terms = 2x
3
+5x
2
-2x +1 Combining like
v
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
-
2
7
y
2
-5 =
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
+
2
7
y
2
+5 =
1
3
y
3
-
2
3
y
3
+
5
7
y
2
+
2
7
y
2
+y -2 +5 Collecting like terms = -
1
3
y
3
+y
2
+y +3 Combining like terms
vi
2
3
x +
3
2
y -
4
3
z -
3
2
x -
5
4
y -
7
2
z =
2
3
x +
3
2
y -
4
3
z -
3
2
x +
5
4
y +
7
2
z =
2
3
x -
3
2
x +
3
2
y +
5
4
y -
4
3
z +
7
2
z Collecting like terms = -
5
6
x +
11
4
y +
13
6
z Combining like terms
vii
2
3
x
2
y +
3
2
xy
2
-
1
3
xy - x
2
y -
4
5
xy
2
+
4
3
xy =
2
3
x
2
y +
3
2
xy
2
-
1
3
xy -x
2
y +
4
5
xy
2
-
4
3
xy =
2
3
x
2
y -x
2
y +
3
2
xy
2
+
4
5
xy
2
-
1
3
xy -
4
3
xy Collecting like terms) = -
1
3
x
2
y +
23
10
xy
2
-
5
3
xy Combining like terms
viii
3
5
bc -
4
5
ac -
ab
7
-
35
3
bc +
6
5
ac =
3
5
bc -
4
5
ac -
ab
7
+
35
3
bc -
6
5
ac =
3
5
bc +
35
3
bc -
4
5
ac -
6
5
ac -
ab
7
Collecting like terms) =
184
15
bc -2ac -
ab
7
Combining like terms
Q u e s t i o n : 5
Take away:
i
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x from
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
ii
5a
2
2
+
3a
3
2
+
a
3
-
6
5
from
1
3
a
3
-
3
4
a
2
-
5
2
iii
7
4
x
3
+
3
5
x
2
+
1
2
x +
9
2
from
7
2
-
x
3
-
x
2
5
iv
y
3
3
+
7
3
y
2
+
1
2
y +
1
2
from
1
3
-
5
3
y
2
v
2
3
ac -
5
7
ab +
2
3
bc from
3
2
ab -
7
4
ac -
5
6
bc
S o l u t i o n :
i The difference is given by:
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x =
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
+
4
5
x
3
-
5
6
-
3
2
x
=
x
3
3
+
4
5
x
3
-
5
2
x
2
-
6
5
x
2
+
3
5
x -
3
2
x +
1
4
-
5
6
Collectingliketerms
=
5+12
15
x
3
+
-25-12
10
x
2
+
6-15
10
x +
6-20
24
=
17
15
x
3
-
37
10
x
2
-
9
10
x -
7
12
Combiningliketerms
ii The difference is given by:
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
+
3a
3
2
+
a
3
-
6
5
=
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
-
3a
3
2
-
a
3
+
6
5
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( (
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( (
( ) ( ) ( ) ( ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
=
1
3
a
3
-
3a
3
2
-
3a
2
4
-
5a
2
2
-
a
3
-
5
2
+
6
5
Collectingliketerms
=
2-9
6
a
3
+
-3-10
4
a
2
-
a
3
+
-25+12
10
= -
7
6
a
3
-
13
4
a
2
-
a
3
-
13
10
Combiningliketerms
iii The difference is given by:
7
2
-
x
3
-
x
2
5
-
7x
3
4
+
3x
2
5
+
x
2
+
9
2
=
7
2
-
x
3
-
x
2
5
-
7x
3
4
-
3x
2
5
-
x
2
-
9
2
=
7
2
-
9
2
-
x
3
-
x
2
-
x
2
5
-
3x
2
5
-
7x
3
4
Collectingliketerms
=
7-9
2
+
-2-3
6
x +
-1-3
5
x
2
-
7x
3
4
= -1 -
5x
6
-
4x
2
5
-
7x
3
4
Combiningliketerms
iv The difference is given by:
1
3
-
5
3
y
2
-
y
3
3
+
7y
2
3
+
y
2
+
1
2
=
1
3
-
5
3
y
2
-
y
3
3
-
7y
2
3
-
y
2
-
1
2
=
1
3
-
1
2
-
y
2
-
5
3
y
2
-
7y
2
3
-
y
3
3
Collectingliketerms
=
2-3
6
-
y
2
+
-5-7
3
y
2
-
y
3
3
= -
1
6
-
y
2
-4y
2
-
y
3
3
Combiningliketerms.
v The difference is given by:
3
2
ab -
7
4
ac -
5
6
bc -
2
3
ac -
5
7
ab +
2
3
bc =
3
2
ab -
7
4
ac -
5
6
bc -
2
3
ac +
5
7
ab -
2
3
bc
=
3
2
ab +
5
7
ab -
7
4
ac -
2
3
ac -
5
6
bc -
2
3
bc Collectingliketerms
=
21+10
14
ab +
-21-8
12
ac +
-5-4
6
bc
=
31
14
ab -
29
12
ac -
3
2
bc Combiningliketerms
Q u e s t i o n : 6
Subtract 3x - 4y - 7z from the sum of x - 3y + 2z and - 4x + 9y - 11z.
S o l u t i o n :
Let first add the expressions x -3y +2z and -4x +9y -11z. We get:
(x -3y +2z)+(-4x +9y -11z)
= x -3y +2z -4x +9y -11z
= x -4x -3y +9y +2z -11z Collectingliketerms
= -3x +6y -9z Combiningliketerms
Now, subtracting the expression 3x -4y -7z from the above sum; we get:
(-3x +6y -9z)-(3x -4y -7z) = -3x +6y -9z -3x +4y +7z
= -3x -3x +6y +4y -9z +7z Collectingliketerms
= -6x +10y -2z Combiningliketerms
Thus, the answer is -6x +10y -2z.
Q u e s t i o n : 7
Subtract the sum of 3l - 4m - 7n
2
and 2l + 3m - 4n
2
from the sum of 9l + 2m - 3n
2
and - 3l + m + 4n
2
.....
S o l u t i o n :
We have to subtract the sum of (3l - 4m - 7n
2
) and (2l + 3m - 4n
2
) from the sum of (9l + 2m - 3n
2
) and (-3l + m + 4n
2
) ? 9l +2m -3n
2
+ -3l +m +4n
2
- 3l -4m -7n
2
+ 2l +3m -4n
2
= 9l -3l +2m +m -3n
2
+4n
2
- 3l +2l -4m +3m -7n
2
-4n
2
= 6l +3m +n
2
- 5l -m -11n
2
Combiningliketermsinsidetheparentheses
= 6l +3m +n
2
-5l +m +11n
2
= 6l -5l +3m +m +n
2
+11n
2
Collectingliketerms
= l +4m +12n
2
Combiningliketerms
Thus, the required solution is l +4m +12n
2
.
Q u e s t i o n : 8
Subtract the sum of 2x - x
2
+ 5 and - 4x - 3 + 7x
2
from 5.
S o l u t i o n :
We have to subtract the sum of (2x - x
2
+ 5) and (- 4x - 3 + 7x) from 5.
5 - 2x -x
2
+5 + -4x -3 +7x
2
= 5 - 2x -4x -x
2
+7x
2
+5 -3 = 5 -2x +4x +x
2
-7x
2
-5 +3
= 5 -5 +3 -2x +4x +x
2
-7x
2
Collectingliketerms
= 3 +2x -6x
2
Combiningliketerms
Thus, the answer is 3 +2x -6x
2
.
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
{ ( ) ( )} { ( ) ( )}
( ) ( )
( ) ( )
{ ( ) ( )} ( )
Page 4
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?
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?
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?
?
?
?
?
?
?
Q u e s t i o n : 3
Add the following algebraic expressions:
i 3a
2
b, - 4a
2
b, 9a
2
b
ii
2
3
a,
3
5
a, -
6
5
a
iii 4xy
2
- 7x
2
y, 12x
2
y - 6xy
2
, - 3x
2
y +5xy
2
iv
3
2
a -
5
4
b +
2
5
c,
2
3
a -
7
2
b +
7
2
c,
5
3
a +
5
2
b -
5
4
c
v
11
2
xy +
12
5
y +
13
7
x, -
11
2
y -
12
5
x -
13
7
xy
vi
7
2
x
3
-
1
2
x
2
+
5
3
,
3
2
x
3
+
7
4
x
2
-x +
1
3
,
3
2
x
2
-
5
2
x -2
S o l u t i o n :
i To add the like terms, we proceed as follows:
3a
2
b + -4a
2
b +9a
2
b = 3a
2
b -4a
2
b +9a
2
b = (3 -4 +9)a
2
b Distributive law = 8a
2
b
ii To add the like terms, we proceed as follows:
2
3
a +
3
5
a + -
6
5
a =
2
3
a +
3
5
a -
6
5
a =
2
3
+
3
5
-
6
5
a Distributive law =
1
15
a
( ) ( )
( ) ( ) ( )
iii To add, we proceed as follows:
4xy
2
-7x
2
y + 12x
2
y + -6xy
2
+ -3x
2
y +5xy
2
= 4xy
2
-7x
2
y +12x
2
y -6xy
2
-3x
2
y +5xy
2
= 4xy
2
-6xy
2
+5xy
2
-7x
2
y +12x
2
y -3x
2
y Collecting like terms = 3xy
2
+2x
2
y
iv To add, we proceed as follows:
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a -
5
4
b +
2
5
c +
2
3
a -
7
2
b +
7
2
c +
5
3
a +
5
2
b -
5
4
c =
3
2
a +
2
3
a +
5
3
a -
5
4
b -
7
2
b +
5
2
b +
2
5
c +
7
2
c -
5
4
c Collecting like terms =
23
6
a -
9
4
b +
53
20
c
v To add, we proceed as follows:
11
2
xy +
12
5
y +
13
7
x + -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy +
12
5
y +
13
7
x -
11
2
y -
12
5
x -
13
7
xy =
11
2
xy -
13
7
xy +
12
5
y -
11
2
y +
13
7
x -
12
5
x Collecting like terms) =
51
14
xy -
31
10
y -
19
35
x Combining like terms)
vi To add, we proceed as follows:
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
-
1
2
x
2
+
5
3
+
3
2
x
3
+
7
4
x
2
-x +
1
3
+
3
2
x
2
-
5
2
x -2 =
7
2
x
3
+
3
2
x
3
-
1
2
x
2
+
7
4
x
2
+
3
2
x
2
-x -
5
2
x +
5
3
+
1
3
-2 Collecting like terms = 5x
3
+
11
4
x
2
-
7
2
x
Q u e s t i o n : 4
Subtract:
i - 5xy from 12xy
ii 2a
2
from - 7a
2
iii 2a - b from 3a - 5b
iv 2x
3
- 4x
2
+ 3x + 5 from 4x
3
+ x
2
+ x + 6
v
2
3
y
3
-
2
7
y
2
-5 from
1
3
y
3
+
5
7
y
2
+y -2
vi
3
2
x -
5
4
y -
7
2
z from
2
3
x +
3
2
y -
4
3
z
vii x
2
y -
4
5
xy
2
+
4
3
xy from
2
3
x
2
y +
3
2
xy
2
-
1
3
xy
viii
ab
7
-
35
3
bc +
6
5
ac from
3
5
bc -
4
5
ac
S o l u t i o n :
(i) 12xy -(-5xy) = 12xy +5xy = 17xy
ii -7a
2
- 2a
2
= -7a
2
-2a
2
= -9a
2
(iii) (3a -5b)-(2a -b) = (3a -5b)-2a +b = 3a -5b -2a +b = 3a -2a -5b +b = a -4b
iv 4x
3
+x
2
+x +6 - 2x
3
-4x
2
+3x +5 = 4x
3
+x
2
+x +6 -2x
3
+4x
2
-3x -5 = 4x
3
-2x
3
+x
2
+4x
2
+x -3x +6 -5 Collecting like terms = 2x
3
+5x
2
-2x +1 Combining like
v
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
-
2
7
y
2
-5 =
1
3
y
3
+
5
7
y
2
+y -2 -
2
3
y
3
+
2
7
y
2
+5 =
1
3
y
3
-
2
3
y
3
+
5
7
y
2
+
2
7
y
2
+y -2 +5 Collecting like terms = -
1
3
y
3
+y
2
+y +3 Combining like terms
vi
2
3
x +
3
2
y -
4
3
z -
3
2
x -
5
4
y -
7
2
z =
2
3
x +
3
2
y -
4
3
z -
3
2
x +
5
4
y +
7
2
z =
2
3
x -
3
2
x +
3
2
y +
5
4
y -
4
3
z +
7
2
z Collecting like terms = -
5
6
x +
11
4
y +
13
6
z Combining like terms
vii
2
3
x
2
y +
3
2
xy
2
-
1
3
xy - x
2
y -
4
5
xy
2
+
4
3
xy =
2
3
x
2
y +
3
2
xy
2
-
1
3
xy -x
2
y +
4
5
xy
2
-
4
3
xy =
2
3
x
2
y -x
2
y +
3
2
xy
2
+
4
5
xy
2
-
1
3
xy -
4
3
xy Collecting like terms) = -
1
3
x
2
y +
23
10
xy
2
-
5
3
xy Combining like terms
viii
3
5
bc -
4
5
ac -
ab
7
-
35
3
bc +
6
5
ac =
3
5
bc -
4
5
ac -
ab
7
+
35
3
bc -
6
5
ac =
3
5
bc +
35
3
bc -
4
5
ac -
6
5
ac -
ab
7
Collecting like terms) =
184
15
bc -2ac -
ab
7
Combining like terms
Q u e s t i o n : 5
Take away:
i
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x from
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
ii
5a
2
2
+
3a
3
2
+
a
3
-
6
5
from
1
3
a
3
-
3
4
a
2
-
5
2
iii
7
4
x
3
+
3
5
x
2
+
1
2
x +
9
2
from
7
2
-
x
3
-
x
2
5
iv
y
3
3
+
7
3
y
2
+
1
2
y +
1
2
from
1
3
-
5
3
y
2
v
2
3
ac -
5
7
ab +
2
3
bc from
3
2
ab -
7
4
ac -
5
6
bc
S o l u t i o n :
i The difference is given by:
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
-
4
5
x
3
+
5
6
+
3
2
x =
x
3
3
-
5
2
x
2
+
3
5
x +
1
4
-
6
5
x
2
+
4
5
x
3
-
5
6
-
3
2
x
=
x
3
3
+
4
5
x
3
-
5
2
x
2
-
6
5
x
2
+
3
5
x -
3
2
x +
1
4
-
5
6
Collectingliketerms
=
5+12
15
x
3
+
-25-12
10
x
2
+
6-15
10
x +
6-20
24
=
17
15
x
3
-
37
10
x
2
-
9
10
x -
7
12
Combiningliketerms
ii The difference is given by:
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
+
3a
3
2
+
a
3
-
6
5
=
1
3
a
3
-
3a
2
4
-
5
2
-
5a
2
2
-
3a
3
2
-
a
3
+
6
5
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( (
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) (
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( (
( ) ( ) ( ) ( ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
=
1
3
a
3
-
3a
3
2
-
3a
2
4
-
5a
2
2
-
a
3
-
5
2
+
6
5
Collectingliketerms
=
2-9
6
a
3
+
-3-10
4
a
2
-
a
3
+
-25+12
10
= -
7
6
a
3
-
13
4
a
2
-
a
3
-
13
10
Combiningliketerms
iii The difference is given by:
7
2
-
x
3
-
x
2
5
-
7x
3
4
+
3x
2
5
+
x
2
+
9
2
=
7
2
-
x
3
-
x
2
5
-
7x
3
4
-
3x
2
5
-
x
2
-
9
2
=
7
2
-
9
2
-
x
3
-
x
2
-
x
2
5
-
3x
2
5
-
7x
3
4
Collectingliketerms
=
7-9
2
+
-2-3
6
x +
-1-3
5
x
2
-
7x
3
4
= -1 -
5x
6
-
4x
2
5
-
7x
3
4
Combiningliketerms
iv The difference is given by:
1
3
-
5
3
y
2
-
y
3
3
+
7y
2
3
+
y
2
+
1
2
=
1
3
-
5
3
y
2
-
y
3
3
-
7y
2
3
-
y
2
-
1
2
=
1
3
-
1
2
-
y
2
-
5
3
y
2
-
7y
2
3
-
y
3
3
Collectingliketerms
=
2-3
6
-
y
2
+
-5-7
3
y
2
-
y
3
3
= -
1
6
-
y
2
-4y
2
-
y
3
3
Combiningliketerms.
v The difference is given by:
3
2
ab -
7
4
ac -
5
6
bc -
2
3
ac -
5
7
ab +
2
3
bc =
3
2
ab -
7
4
ac -
5
6
bc -
2
3
ac +
5
7
ab -
2
3
bc
=
3
2
ab +
5
7
ab -
7
4
ac -
2
3
ac -
5
6
bc -
2
3
bc Collectingliketerms
=
21+10
14
ab +
-21-8
12
ac +
-5-4
6
bc
=
31
14
ab -
29
12
ac -
3
2
bc Combiningliketerms
Q u e s t i o n : 6
Subtract 3x - 4y - 7z from the sum of x - 3y + 2z and - 4x + 9y - 11z.
S o l u t i o n :
Let first add the expressions x -3y +2z and -4x +9y -11z. We get:
(x -3y +2z)+(-4x +9y -11z)
= x -3y +2z -4x +9y -11z
= x -4x -3y +9y +2z -11z Collectingliketerms
= -3x +6y -9z Combiningliketerms
Now, subtracting the expression 3x -4y -7z from the above sum; we get:
(-3x +6y -9z)-(3x -4y -7z) = -3x +6y -9z -3x +4y +7z
= -3x -3x +6y +4y -9z +7z Collectingliketerms
= -6x +10y -2z Combiningliketerms
Thus, the answer is -6x +10y -2z.
Q u e s t i o n : 7
Subtract the sum of 3l - 4m - 7n
2
and 2l + 3m - 4n
2
from the sum of 9l + 2m - 3n
2
and - 3l + m + 4n
2
.....
S o l u t i o n :
We have to subtract the sum of (3l - 4m - 7n
2
) and (2l + 3m - 4n
2
) from the sum of (9l + 2m - 3n
2
) and (-3l + m + 4n
2
) ? 9l +2m -3n
2
+ -3l +m +4n
2
- 3l -4m -7n
2
+ 2l +3m -4n
2
= 9l -3l +2m +m -3n
2
+4n
2
- 3l +2l -4m +3m -7n
2
-4n
2
= 6l +3m +n
2
- 5l -m -11n
2
Combiningliketermsinsidetheparentheses
= 6l +3m +n
2
-5l +m +11n
2
= 6l -5l +3m +m +n
2
+11n
2
Collectingliketerms
= l +4m +12n
2
Combiningliketerms
Thus, the required solution is l +4m +12n
2
.
Q u e s t i o n : 8
Subtract the sum of 2x - x
2
+ 5 and - 4x - 3 + 7x
2
from 5.
S o l u t i o n :
We have to subtract the sum of (2x - x
2
+ 5) and (- 4x - 3 + 7x) from 5.
5 - 2x -x
2
+5 + -4x -3 +7x
2
= 5 - 2x -4x -x
2
+7x
2
+5 -3 = 5 -2x +4x +x
2
-7x
2
-5 +3
= 5 -5 +3 -2x +4x +x
2
-7x
2
Collectingliketerms
= 3 +2x -6x
2
Combiningliketerms
Thus, the answer is 3 +2x -6x
2
.
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{ ( ) ( )} ( )
Q u e s t i o n : 9
Simplify each of the following:
i x
2
- 3x + 5 -
1
2
(3x
2
- 5x + 7)
ii [5 - 3x + 2y - (2x - y)] - (3x - 7y + 9)
iii
11
2
x
2
y -
9
4
xy
2
+
1
4
xy -
1
14
y
2
x +
1
15
yx
2
+
1
2
xy
iv
1
3
y
2
-
4
7
y +11 -
1
7
y -3 +2y
2
-
2
7
y -
2
3
y
2
+2
v -
1
2
a
2
b
2
c +
1
3
ab
2
c -
1
4
abc
2
-
1
5
cb
2
a
2
+
1
6
cb
2
a -
1
7
c
2
ab +
1
8
ca
2
b.
S o l u t i o n :
i x
2
-3x +5 -
1
2
3x
2
-5x +7 = x
2
-3x +5 -
3x
2
2
+
5x
2
-
7
2
= x
2
-
3x
2
2
-3x +
5x
2
+5 -
7
2
Collecting like terms =
1-3
2
x
2
+
-3+5
2
x +
10-7
2
= -
x
2
2
-
x
2
+
3
2
Thus, the answer is -
x
2
2
-
x
2
+
3
2
.
(ii) [5 -3x +2y -(2x -y)] -(3x -7y +9) = [5 -3x +2y -2x +y] -(3x -7y +9) = [5 -5x +3y] -(3x -7y +9) = 5 -5x +3y -3x +7y -9 = 5 -9 -5x -3x +3y +7y = -4 -8x +10y
iii
11
2
x
2
y -
9
4
xy
2
+
1
4
xy -
1
14
y
2
x +
1
15
yx
2
+
1
2
xy
=
11
2
x
2
y +
1
15
yx
2
-
9
4
xy
2
-
1
14
y
2
x +
1
4
xy +
1
2
xy Collectingliketerms
=
165+2
30
x
2
y +
-63-2
28
xy
2
+
1+2
4
xy
=
167
30
x
2
y -
65
28
y
2
x +
3
2
xy Combiningliketerms
iv
1
3
y
2
-
4
7
y +11 -
1
7
y -3 +2y
2
-
2
7
y -
2
3
y
2
+2 =
1
3
y
2
-
4
7
y +11 -
1
7
y +3 -2y
2
-
2
7
y +
2
3
y
2
-2
=
1
3
y
2
-2y
2
+
2
3
y
2
-
4
7
y -
1
7
y -
2
7
y +11 +3 -2 Collectingliketerms
=
1-6+2
3
y
2
+
-4-1-2
7
y +12
= -y
2
-7y +12 Combiningliketerms
v -
1
2
a
2
b
2
c +
1
3
ab
2
c -
1
4
abc
2
-
1
5
cb
2
a
2
+
1
6
cb
2
a -
1
7
c
2
ab +
1
8
ca
2
b
= -
1
2
a
2
b
2
c -
1
5
cb
2
a
2
+
1
3
ab
2
c +
1
6
cb
2
a -
1
4
abc
2
-
1
7
c
2
ab +
1
8
ca
2
b Collectingliketerms
=
-5-2
10
a
2
b
2
c +
2+1
6
cb
2
a
2
+
-7-4
28
c
2
ab +
1
8
ca
2
b
= -
7
10
a
2
b
2
c +
1
2
ab
2
c -
11
28
abc
2
+
1
8
a
2
bc Combiningliketerms
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
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