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### Access answers to Chapter 8 – Algebraic Expressions

## Exercise 8A PAGE NO: 130

**1. Write the following using literals, numbers and signs of basic operation:**

**(i) x increased by 12**

**(ii) y decreased by 7**

**(iii) The difference of a and b, when a > b**

**(iv) The product of x and y added to their sum**

**(v) One third of x multiplied by the sum of a and b**

**(vi) 5 times x added to seven times y**

**(vii) Sum of x and the quotient of y by 5**

**(viii) x taken away from 4**

**(ix) 2 less than the quotient of x by y**

**(x) x multiplied by itself**

**(xi) Twice x increased by y**

**(xii) Thrice x added to y squared**

**(xiii) x minus twice y**

**(xiv) x cubed less than y cubed**

**(xv) The quotient of x by 8 is multiplied by y**

**Solution**

(i) x increased by 12 is written as x + 12

(ii) y decreased by 7 is written as y â€“ 7

(iii) The difference of a and b, when a > b is (a â€“ b)

(iv) The product of x and y is xy, added to their sum is (x + y)

Hence, the product of x and y added to their sum is (x + y) + xy

(v) One third of x = x / 3

Sum of a and b = (a + b)

âˆ´ One third of x multiplied by the sum of a and b = x / 3 Ã— (a + b)

= x (a + b) / 3

(vi) 5 times x = 5x, seven times y = 7y

âˆ´ 5 times x added to seven times y is written as 7y + 5x

(vii) Sum of x = x

Quotient of y = y / 5

âˆ´ Sum of x and the quotient of y by 5 = x + y / 5

(viii) x taken away from 4 is written as (4 â€“ x)

(ix) Quotient of x by y = x / y

âˆ´ 2 less than the quotient of x by y = (x / y) â€“ 2

(x) x multiplied by itself is x Ã— x = x^{2}

âˆ´ x multiplied by itself is written as x^{2}

(xi) Twice x increased by y is written as (2x + y)

(xii) Thrice x = 3 Ã— x = 3x and y squared = (y Ã— y) = y^{2}

âˆ´ Thrice x added to y squared is written as 3x + y^{2}

(xiii) Twice y = 2 Ã— y = 2y

âˆ´ x minus twice y is written as (x â€“ 2y)

(xiv) x cubed = (x Ã— x Ã— x) = x^{3 }y cubed = (y Ã— y Ã— y ) = y^{3}

^{ }âˆ´ x cubed less than y cubed is written as (y^{3 }– x^{3})

(xv) The quotient of x by 8 is x / 8

âˆ´ The quotient of x by 8 is multiplied by y is written as (x / 8) Ã— y

**2. Ranjit scores 80 marks in English and x marks in Hindi. What is his total score in the two subjects?**

**Solution**

Marks scored by Ranjit in English = 80

Marks scored by Ranjit in Hindi = x

Total score in the two subjects = (Marks in English) + (Marks in Hindi)

= (80 + x)

âˆ´ Total score in the two subjects scored by Ranjit is (80 + x) marks

## Exercise 8B PAGE NO: 132

**1. If a = 2 and b = 3, find the value of**

**(i) a + b**

**(ii) a ^{2 }+ ab**

**(iii) ab â€“ a ^{2 }**

**(iv) 2a â€“ 3b**

**(v) 5a ^{2} â€“ 2ab**

**(vi) a ^{3} â€“ b^{3}**

**Solutions**

Given a = 2 and b = 3

(i) a + b

Substituting a = 2 and b = 3 in the given expression, we get

a + b = 2 + 3

= 5

** **âˆ´ a + b = 5

(ii) a^{2} + ab

Substituting a = 2 and b = 3 in the given expression, we get

** **a^{2} + ab = (2)^{2 }+ (2) Ã— (3)

= (2 Ã— 2) + (2 Ã— 3)

= 4 + 6

= 10

âˆ´ a^{2} + ab = 10

(iii) ab â€“ a^{2}

^{ } Substituting a = 2 and b = 3 in the given expression, we get

ab â€“ a^{2 }= (2 Ã— 3) â€“ (2)^{2}

^{ } = (2 Ã— 3) â€“ (2 Ã— 2)

= 6 â€“ 4

= 2

âˆ´ ab â€“ a^{2 } = 2

(iv) 2a â€“ 3b

Substituting a = 2 and b = 3 in the given expression, we get

2a â€“ 3b = (2 Ã— 2) â€“ (3 Ã— 3)

= 4 â€“ 9

= -5

âˆ´ 2a â€“ 3b = -5

(v) 5a^{2 }â€“ 2ab

Substituting a = 2 and b = 3 in the given expression, we get

5a^{2 }â€“ 2ab = 5 Ã— (2)^{2} â€“ 2 (2) (3)

= (5 4) â€“ 2 (2 Ã— 3)

= 20 â€“ 2 (6)

= 20 â€“ 12

= 8

âˆ´ 5a^{2 }â€“ 2ab = 8

(vi) a^{3 }– b^{3}

Substituting a = 2 and b = 3 in the given expression, we get

a^{3 }– b^{3} = (2)^{3} â€“ (3)^{3}

= (2 Ã— 2 Ã— 2) â€“ (3 Ã— 3 Ã— 3)

= 8 â€“ 27

=-19

âˆ´ a^{3 }– b^{3} = -19

^{ }**2. If x = 1, y = 2 and z = 5, find the value of**

**(i) 3x â€“ 2y + 4z**

**(ii) x ^{2 }+ y^{2} + z^{2}**

**(iii) ^{ }2x^{2 }– 3 y^{2} + z^{2}**

**(iv) xy + yz -zx**

**(v) 2x ^{2}y â€“ 5yz + xy^{2}**

**(vi) x ^{3 }– y^{3} – z^{3}**

**Solutions**

Given x = 1, y = 2 and z = 5

(i) 3x â€“ 2y + 4z

Substituting x = 1, y = 2 and z = 5 in the given expression, we get

3x â€“ 2y + 4z = 3 (1) â€“ 2 (2) + 4 (5)

= (3 Ã— 1) â€“ (2 Ã— 2) + (4 Ã— 5)

= 3 â€“ 4 + 20

= 23 â€“ 4

= 19

âˆ´ 3x â€“ 2y + 4z = 19

(ii) x^{2 }+ y^{2} + z^{2}

^{ }Substituting x = 1, y = 2 and z = 5 in the given expression, we get

^{ }x^{2 }+ y^{2} + z^{2} = (1)^{2} + (2)^{2} + (5)^{2}

^{ }= (1Ã— 1) + (2 Ã— 2) + (5 Ã— 5)

= 1 + 4 + 25

= 30

âˆ´ x^{2 }+ y^{2} + z^{2} = 30

(iii) 2x^{2 }â€“ 3y^{2 }+ z^{2}

^{ } Substituting x = 1, y = 2 and z = 5 in the given expression, we get

2x^{2 }â€“ 3y^{2 }+ z^{2} = 2 (1)^{2 }â€“ 3(2)^{2} + (5)^{2}

^{ }= (2 Ã— 1) â€“ 3 (2 Ã— 2) + (5 Ã— 5)

= (2) â€“ 3 (4) + (25)

= 2 â€“ 12 + 25

= 27 â€“ 12

= 15

âˆ´ 2x^{2 }â€“ 3y^{2 }+ z^{2} = 15

(iv) xy + yz â€“zx

^{ } Substituting x = 1, y = 2 and z = 5 in the given expression, we get

xy + yz â€“zx = (1) (2) + (2) (5) â€“ (5) (1)

= (1 Ã— 2) + (2 Ã— 5) â€“ (5 Ã— 1)

= 2 + 10 -5

= 12 â€“ 5

= 7

âˆ´ xy + yz â€“zx = 7

(v) 2x^{2}y â€“ 5yz + xy^{2}

Substituting x = 1, y = 2 and z = 5 in the given expression, we get

2x^{2}y â€“ 5yz + xy^{2} = 2 (1)^{2 }(2) â€“ 5(2) (5) + (1) (2)^{2}

= 2 (1 Ã— 1) (2) â€“ 5 (2 Ã— 5) + (1 Ã— 2 Ã— 2)

= 4 â€“ 5 (10) + (4)

= 4 â€“ 50 + 4

= 8 â€“ 50

= – 42

âˆ´ 2x^{2}y â€“ 5yz + xy^{2} = -42

(vi) x^{3 }– y^{3} – z^{3}

Substituting x = 1, y = 2 and z = 5 in the given expression, we get

x^{3 }– y^{3} – z^{3} = (1)^{3 }â€“ (2)^{3 }â€“ (5)^{3}

^{ } = (1 Ã— 1 Ã— 1) â€“ (2 Ã— 2 Ã— 2) â€“ (5 Ã— 5 Ã— 5)

= (1) â€“ (8) â€“ (125)

= 1 â€“ 8 â€“ 125

= 1 â€“ 133

= -132

âˆ´ x^{3 }– y^{3} – z^{3} = -132

**3. If p = -2, q = -1 and r = 3, find the value of**

**(i) p ^{2 }+ q^{2} – r^{2}**

**(ii) 2p ^{2 }– q^{2} + 3r^{2}**

**(iii) ^{ }p â€“ q – r **

**(iv) p ^{3} + q^{3} + r^{3 }+ 3pqr**

**(v) 3p ^{2}q + 5pq^{2} + 2pqr **

**(vi) p ^{4 }+ q^{4} â€“ r^{4}**

**Solutions**

** **Given p = -2, q = -1 and r = 3

(i) p^{2 }+ q^{2 }â€“ r^{2}

^{ }Substituting p = -2, q = -1 and r = 3 in the given expression, we get

p^{2 }+ q^{2 }â€“ r^{2} = (-2)^{2} + (-1)^{2 }â€“ (3)^{2}

^{ }= (-2 Ã— -2) + (-1 Ã— -1) â€“ (3 Ã— 3)

= 4 + 1 â€“ 9

= 5 â€“ 9

= -4

âˆ´ p^{2 }+ q^{2 }â€“ r^{2} = -4

(ii) 2p^{2 }– q^{2} + 3r^{2}

Substituting p = -2, q = -1 and r = 3 in the given expression, we get

2p^{2 }– q^{2} + 3r^{2} = 2 (-2)^{2} â€“ (-1)^{2 }+ 3 (3)^{2}

^{ } = 2 (-2 Ã— -2) â€“ (-1 Ã— -1) + 3 (3 Ã— 3)

= 2 (4) â€“ (1) + 3 (9)

= 8 â€“ 1 + 27

= 7 + 27

= 34

âˆ´ 2p^{2 }– q^{2} + 3r^{2} = 34

(iii) p â€“ q – r

Substituting p = -2, q = -1 and r = 3 in the given expression, we get

p â€“ q â€“ r = (-2) â€“ (-1) â€“ (3)

= -2 + 1 â€“ 3

= -1 â€“ 3

= -4

âˆ´ p â€“ q â€“ r = -4

(iv) p^{3} + q^{3} + r^{3 }+ 3pqr

Substituting p = -2, q = -1 and r = 3 in the given expression, we get

p^{3} + q^{3} + r^{3 }+ 3pqr = (-2)^{3 }+ (-1)^{3 }+ (3)^{3} + 3(-2) (-1) (3)

= (-2 Ã— -2 Ã— -2) + (-1 Ã— -1 Ã— -1) + (3 Ã— 3 Ã— 3) + 3 (-2 Ã— -1 Ã— 3)

= -8 -1 + 27 + 18

= -9 + 45

= 36

âˆ´ p^{3} + q^{3} + r^{3 }+ 3pqr = 36

(v) 3p^{2}q + 5pq^{2} + 2pqr** **

** **Substituting p = -2, q = -1 and r = 3 in the given expression, we get

** **3p^{2}q + 5pq^{2} + 2pqr** = **3** (-**2)** ^{2} **(-1) + 5 (-2) (-1)

^{2}+ 2 (-2) (-1) (3)

= 3 (-2 Ã— -2 Ã— -1) + 5 (-2 Ã— -1 Ã— -1) + 2 (-2 Ã— -1 Ã— 3)

= 3(-4) + 5(-2) + 2 (6)

= -12 – 10 + 12

= -10

âˆ´ 3p^{2}q + 5pq^{2} + 2pqr** = **-10

(vi) p^{4 }+ q^{4} â€“ r^{4}

Substituting p = -2, q = -1 and r = 3 in the given expression, we get

** **p^{4 }+ q^{4} â€“ r^{4} = (-2)^{4} + (-1)^{4} â€“ (3)^{4}

^{ } = (-2 Ã— -2 Ã— -2 Ã— -2) + (-1 Ã— -1 Ã— -1 Ã— -1) â€“ (3 Ã— 3 Ã— 3 Ã— 3)

= (16) + (1) â€“ (81)

= 17 â€“ 81

= -64

âˆ´ p^{4 }+ q^{4} â€“ r^{4} = -64

**4. Write the coefficient of**

**(i) x in 13 x**

**(ii) y in â€“ 5y**

**(iii) a in 6ab**

**(iv) z in -7xz**

**(v) p in â€“ 2pqr**

**(vi) y ^{2} in 8xy^{2}z**

**(vii) x ^{3} in x^{3}**

**(viii) x ^{2} in – x^{2}**

**Solution**

**(i) **The coefficient of x in 13x is 13

(ii) The coefficient of y in -5y is -5

(iii) The coefficient of a in 6ab is 6b

(iv) The coefficient of z in -7xz is -7x

(v) The coefficient of p in -2pqr is -2qr

(vi) The coefficient of y^{2 }in 8xy^{2}z is 8xz

(vii) The coefficient of x^{3 }in x^{3 }is 1

(viii) The coefficient of x^{2 }in -x^{2} is -1

**5. Write the numerical coefficient of**

**(i) ab**

**(ii) – 6bc**

**(iii) 7xyz**

**(iv) – 2x ^{3} y^{2} z**

**Solutions**

**(i) **The numerical coefficient of ab is 1

(ii) The numerical coefficient -6bc is -6

(iii) The numerical coefficient 7xyz is 7

(iv) The numerical coefficient -2x^{3}y^{2}z is -2

## Exercise 8C PAGE NO: 134

**1. Add: **

**(i) 3x, 7x**

**(ii) 7y, -9y**

**(iii) 2xy, 5xy, -xy**

**(iv) 3x, 2y**

**(v) 2x ^{2}, -3x^{2}, 7x^{2}**

**(vi) 7xyz, -5xyz, 9xyz, -8xyz**

**(vii) 6a ^{3}, -4a^{3}, 10a^{3}, -8a^{3}**

**(viii) x ^{2} â€“ a^{2}, – 5x^{2} + 2a^{2}, -4x^{2} + 4a^{2}**

**Solutions**

**(i) **The required sum = 3x + 7x

= (3 + 7) x

= 10x

(ii) The required sum = 7y + (-9y)

= 7y â€“ 9y

= (7 -9) y

= -2y

(iii) The required sum = 2xy + 5xy + (-xy)

= 2xy + 5xy â€“ xy

= (2x + 5x â€“x) y

= 6xy

(iv) The required sum = 3x + 2y

= 3x + 2y

(v) The required sum = 2x^{2} + (-3x^{2}) + 7x^{2}

^{ } = 2x^{2} – 3x^{2} + 7x^{2}

= (2 – 3 + 7) x^{2}

= 6x^{2}

(vi) The required sum = 7xyz + (-5xyz) + 9xyz + (-8xyz)

= 7xyz â€“ 5xyz + 9xyz â€“ 8xyz

= (7 â€“ 5 + 9 â€“ 8) xyz

= (16 â€“ 13) xyz

= 3xyz

(vii) The required sum = 6a^{3}** ^{ }**+ (-4a

^{3}) + 10a

^{3 }+ (-8a

^{3})

= 6a^{3 }â€“ 4a^{3 }+ 10a^{3} â€“ 8a^{3}

= (6 â€“ 4 + 10 â€“ 8) a^{3}

^{ }= (16 â€“ 12) a^{3}

= 4a^{3}

(viii) The required sum = (x^{2 }â€“ a^{2})^{ }+ (-5x^{2} + 2a^{2}) + (-4x^{2 }+ 4a^{2})

= x^{2}– a^{2} â€“ 5x^{2 }+ 2a^{2} â€“ 4x^{2 }+ 4a^{2}

= (1 â€“ 5 â€“ 4) x^{2 }â€“ (1 â€“ 2 â€“ 4) a^{2}

= (1 â€“ 9) x^{2 }â€“ (1 â€“ 6) a^{2}

^{ }= -8x^{2 }+ 5a^{2}

**2. Add the following:**

**(i) x â€“ 3y â€“ 2z (ii) m ^{2} â€“ 4m + 5 (iii) 2x^{2} â€“ 3xy + y^{2} **

** 5x + 7y â€“ z -2m ^{2} + 6m â€“ 6 – 7x^{2} â€“ 5xy â€“ 2y^{2}**

** -7x â€“ 2y + 4z -m ^{2} â€“ 2m â€“ 7 4x^{2} + xy â€“ 6y^{2}**

** __________ __________ ____________**

** __________ __________ ____________**

**(iv) 4xy â€“ 5yz â€“ 7z**

** – 5xy +2yz + zx**

** – 2xy -3yz +3zx**

** ____________**

** ______________**

**Solutions**

**(i) **x â€“ 3y â€“ 2z

5x + 7y â€“ z

-7x â€“ 2y + 4z

____________________

-x + 2y + z

____________________

(ii) m ^{2}â€“ 4m + 5

-2m^{2 }+ 6m â€“ 6

– m^{2 }– 2m â€“ 7

_______________

-2m^{2} +0m â€“ 8

= -2m^{2} – 8

________________

(iii) 2x^{2 }â€“ 3xy + y^{2}

^{ } -7x^{2 }â€“ 5xy â€“ 2y^{2}

^{ } 4x^{2} + xy â€“ 6y^{2}

^{ ___________________________}

^{ } -x^{2 }â€“ 7xy â€“ 7y^{2}

^{ _____________________________}

(iv) 4xy â€“ 5yz â€“ 7zx

-5xy + 2yz + zx

-2xy â€“ 3yz + 3zx

_________________

-3xy -6yz -3zx

________________

**3. Add:**

**(i) 3a â€“ 2b + 5c, 2a +5b â€“ 7c, – a â€“ b + C**

**(ii) 8a â€“ 6ab + 5b, – 6a â€“ ab â€“ 8b, – 4a + 2ab + 3b**

**(iii) 2x ^{3} â€“ 3x^{2} + 7x â€“ 8, – 5x^{3} + 2x^{2} â€“ 4x + 1, 3 â€“ 6x + 5x^{2} â€“ x^{3}**

**(iv) 2x ^{2} â€“ 8xy + 7y^{2} â€“ 8xy^{2}, 2xy^{2} + 6xy â€“ y^{2} + 3x^{2}, 4y^{2} â€“ xy â€“ x^{2} + xy^{2}**

**(v) x ^{3} + y^{3} – z^{3} + 3xyz, -x^{3} + y^{3} + z^{3} â€“ 6xyz, x^{3} â€“ y^{3} â€“ z^{3} â€“ 8xyz**

**(vi) 2 + x â€“ x ^{2} + 6x^{3}, – 6 â€“ 2x + 4x^{2} â€“ 3x^{3}, 2 + x^{2}, 3-x^{3} + 4x â€“ 2x^{2}**

**Solution**

**(i) **The sum of the given expressions

= (3a + 2a â€“a) + (-2b +5b â€“b) + (5c â€“ 7c +c)

= 4a + 2b â€“ c

(ii) The sum of the given expressions

= (8a -6a -4a) + (5b â€“ 8b + 3b) + (-6ab â€“ab + 2ab)

= -2a â€“ 5ab

(iii) The sum of the given expressions

= (2x^{3} â€“ 5x^{3} â€“ x^{3}) + (-3x^{2} + 2x^{2} + 5x^{2}) + (7x – 4x â€“ 6x) + (-8 + 1 +3)

= – 4x^{3 }+ 4x^{2 }â€“ 3x – 4

(iv) The sum of the given expressions

** = **(2x^{2 }+ 3x^{2 }â€“ x^{2}) + (-8xy + 6xy â€“ xy) + (7y^{2 }â€“ y^{2 }+ 4y^{2}) + (-8xy^{2} + 2xy^{2 }+ xy^{2})

= 4x^{2} â€“ 3xy + 10 y^{2 }â€“ 5xy^{2}

(v) The sum of the given expressions

= (x^{3 }– x^{3 }+ x^{3}) + (y^{3} + y^{3} â€“ y^{3}) + (-z^{3} + z^{3 }â€“ z^{3}) + (3xyz â€“ 6xyz â€“ 8xyz)

= x^{3} + y^{3} â€“ z^{3} â€“ 11xyz

(vi) The sum of the given expressions

= (2 â€“ 6 + 2 + 3) + (x â€“ 2x + 4x) + (- x^{2} + 4x^{2} + x^{2} â€“ 2x^{2}) + (6x^{3 }â€“ 3x^{3} â€“ x^{3})

= 1 + 3x + 2x^{2} + 2x^{3}

**4. Subtract:**

**(i) 5x from 2x**

**(ii) – xy from 6xy**

**(iii) 3a from 5b**

**(iv) – 7x from 9y**

**(v) 10x ^{2} from – 7x^{2}**

**(vi) a ^{2} â€“ b^{2} from b^{2} â€“ a^{2}**

**Solutions**

**(i) **Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = 5x

Changing the sign of each term of expression = -5x

2x â€“ 5x = (2 â€“ 5) x

= – 3x

(ii) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = -xy

Changing the sign of each term of expression = xy

6xy + xy = (6 + 1) xy

= 7xy

(iii) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = 3a

Changing the sign of each term of expression = – 3a

= (5b â€“ 3a)

(iv) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = – 7x

Changing the sign of each term of expression = 7x

= (9y + 7x)

(v) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = 10x^{2}

Changing the sign of each term of expression = – 10x^{2}

^{ }= (- 7x^{2} â€“ 10x^{2})

= – 17x^{2}

(vi) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

We get

Term which is subtracted = a^{2} â€“ b^{2}

Changing the sign of each term of expression = – (a^{2} â€“ b^{2})

= (b^{2 }â€“ a^{2}) â€“ (a^{2 }â€“ b^{2})

= (b^{2} â€“ a^{2} â€“ a^{2} + b^{2})

= 2b^{2 }â€“ 2a^{2}

**5. Subtract: **

**(i) 5a + 7b â€“ 2c from 3a â€“ 7b + 4c**

**(ii) a â€“ 2b -3c from -2a + 5b â€“ 4c**

**(iii) 5x ^{2} â€“ 3xy + y^{2} from 7x^{2} â€“ 2xy â€“ 4y^{2}**

**(iv) 6x ^{3} â€“ 7x^{2} + 5x â€“ 3 from 4 – 5x + 6x^{2} â€“ 8x^{3}**

**(v) x ^{3} + 2x^{2}y + 6xy^{2} â€“ y^{3} from y^{3} â€“ 3xy^{2} â€“ 4x^{2}y**

**(vi) – 11x ^{2}y^{2} + 7xy â€“ 6 from 9x^{2}y^{2} â€“ 6xy + 9**

**(vii) -2a + b + 6d from 5a -2b -3c**

**Solutions**

**(i) **Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = 5a + 7b – 2c

Changing the sign of each term of expression = – 5a â€“ 7b + 2c

Now add

= (3a â€“ 7b + 4c) + (- 5a – 7b + 2c)

= 3a â€“ 5a â€“ 7b – 7b + 4c + 2c

= -2a â€“ 14b + 6c

(ii) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = a â€“ 2b â€“ 3c

Changing the sign of each term of expression = -a + 2b + 3c

Now add

= (- 2a + 5b – 4c) + (-a + 2b + 3c)

= -2a + 5b â€“ 4c â€“ a + 2b + 3c

= -3a + 7b â€“ c

(iii) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = 5x^{2} â€“ 3xy + y^{2}

Changing the sign of each term of expression = – 5x^{2 }+ 3xy â€“ y^{2}

Now add

= (7x^{2 }– 2xy â€“ 4y^{2}) + (- 5x^{2} + 3xy â€“ y^{2})

= 7x^{2} â€“ 2xy â€“ 4y^{2} â€“ 5x^{2 }+ 3xy â€“ y^{2}

= 2x^{2} + xy â€“ 5y^{2}

(iv) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = 6x^{3} – 7x^{2} + 5x – 3

Changing the sign of each term of expression = -6x^{3} + 7x^{2} â€“ 5x + 3

Now add

= (4 â€“ 5x + 6x^{2} â€“ 8x^{3}) + (- 6x^{3} + 7x^{2} â€“ 5x + 3)

= 4 â€“ 5x + 6x^{2} â€“ 8x^{3 }â€“ 6x^{3} + 7x^{2} â€“ 5x + 3

= 4 + 3 â€“ 5x â€“ 5x + 6x^{2}_{ }+ 7x^{2} â€“ 8x^{3} â€“ 6x^{3}

= 7 â€“ 10x + 13x^{2} â€“ 14x^{3}

(v) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = x^{3} + 2x^{2}y + 6xy^{2} â€“ y^{3}

Changing the sign of each term of expression = -x^{3} â€“ 2x^{2}y â€“ 6xy^{2} + y^{3}

Now add

= (y^{3} â€“ 3xy^{2} â€“ 4x^{2}y) + (- x^{3} â€“ 2x^{2}y â€“ 6xy^{2} + y^{3})

= y^{3} â€“ 3xy^{2} â€“ 4x^{2}y â€“ x^{3} â€“ 2x^{2}y â€“ 6xy^{2} + y^{3}

= y^{3} + y^{3} â€“ 3xy^{2} â€“ 6xy^{2} â€“ 4x^{2}y â€“ 2x^{2}y â€“ x^{3}

^{ }= 2y^{3} â€“ 9xy^{2} â€“ 6x^{2}y â€“ x^{3}

(vi) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = -11x^{2}y^{2} + 7xy – 6

Changing the sign of each term of expression = 11x^{2}y^{2 }â€“ 7xy + 6

Now add

= (9x^{2}y^{2} â€“ 6xy + 9) + (11x^{2}y^{2} â€“ 7xy + 6)

= 9x^{2}y^{2} â€“ 6xy + 9 + 11x^{2}y^{2} â€“ 7xy + 6

= 9x^{2}y^{2} + 11x^{2}y^{2} â€“ 6xy â€“ 7xy + 9 + 6

= 20x^{2}y^{2} â€“ 13xy + 15

(vii) Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = -2a + b + 6d

Changing the sign of each term of expression = 2a â€“ b â€“ 6d

Now add

= (5a â€“ 2b â€“ 3c) + (2a â€“ b â€“ 6d)

= 5a â€“ 2b â€“ 3c + 2a â€“ b â€“ 6d

= 5a + 2a â€“ 2b â€“ b â€“ 3c â€“ 6d

= 7a â€“ 3b â€“ 3c â€“ 6d

** **

**6. Simplify:**

**(i) 2p ^{3} â€“ 3p^{2} + 4p â€“ 5 â€“ 6p^{3} + 2p^{2} â€“ 8p -2 + 6p +8**

**(ii) 2x ^{2} â€“ xy + 6x â€“ 4y + 5xy â€“ 4x + 6x^{2} + 3y**

**(iii) x ^{4} â€“ 6x^{3} + 2x â€“ 7 + 7x^{3} â€“ x + 5x^{2} + 2 â€“ x^{4}**

**Solution**

**(i) **Given

2p^{3} â€“ 3p^{2} + 4p â€“ 5 â€“ 6p^{3} + 2p^{2} â€“ 8p â€“ 2 + 6p + 8

** **Rearranging and collecting the like terms, we get:

= 2p^{3} â€“ 6p^{3} â€“ 3p^{2} + 2p^{2} + 4p â€“ 8p + 6p â€“ 5 + 2 + 8

= (2 â€“ 6) p3 â€“ (3 â€“ 2) p2 + (4 â€“ 8 + 6) p â€“ (5 – 2 â€“ 8)

= (- 4) p^{3} â€“ (1) p^{2} + (10 â€“ 8) p â€“ (7 â€“ 8)

= (-4) p^{3} â€“ p^{2} + (2) p â€“ (-1)

= – 4p^{3 }â€“ p^{2} + 2p + 1

(ii) Given

2x^{2} â€“ xy + 6x â€“ 4y + 5xy â€“ 4x + 6x^{2} + 3y

Rearranging and collecting the like terms, we get:

= 2x^{2} + 6x^{2} â€“ xy + 5xy + 6x â€“ 4x â€“ 4y + 3y

= (2 + 6) x^{2} â€“ (1 â€“ 5) xy + (6 â€“ 4) x â€“ (4 â€“ 3) y

= (8) x^{2} â€“ (- 4) xy + (2) x â€“ (1) y

= 8x^{2} + 4xy + 2x â€“ y

(iii) Given

x^{4} â€“ 6x^{3} + 2x â€“ 7 + 7x^{3} â€“ x + 5x^{2} + 2 â€“ x^{4}

Rearranging and collecting the like terms, we get:

= x^{4} – x^{4} â€“ 6x^{3} + 7x^{3} + 5x^{2} + 2x â€“ x â€“ 7 + 2

= (1 â€“ 1) x^{4} â€“ (6 â€“ 7) x^{3} + 5x^{2} + (2 â€“ 1) x – 7 + 2

= – (-1) x^{3} + 5x^{2} + (1) x â€“ 7 + 2

= x^{3} + 5x^{2} + x â€“ 5

**7. From the sum of 3x ^{2} â€“ 5x + 2 and -5x^{2} â€“ 8x + 6, subtract 4x^{2} â€“ 9x + 7.**

**Solution**

** **To find the sum

Add 3x^{2} â€“ 5x + 2 and -5x^{2} â€“ 8x + 6

(3x^{2} â€“ 5x + 2) + (-5x^{2} â€“ 8x + 6)

Rearranging and collecting the like terms, we get:

= 3x^{2} â€“ 5x^{2} â€“ 5x â€“ 8x + 2 + 6

= (3 â€“ 5) x^{2} â€“ (5 + 8) x + 2 + 6

= – 2x^{2 }â€“ 13x + 8

Now Subtract 4x^{2} â€“ 9x + 7 from -2x^{2 }– 13x + 8

Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = 4x^{2} â€“ 9x + 7

Changing the sign of each term of expression = – 4x^{2} + 9x â€“ 7

= – 2x^{2} â€“ 13x + 8 â€“ 4x^{2} + 9x â€“ 7

= -2x^{2} â€“ 13x + 8 – 4x^{2} â€“ 9x â€“ 7

= -2x^{2 }– 4x^{2 }â€“ 13x â€“ 9x + 8 â€“ 7

= (-2 – 4) x^{2 }â€“ (13 + 9) x + 8 â€“ 7

= – 6x^{2} â€“ 4x + 1

**8. If A = 7x ^{2} + 5xy â€“ 9y^{2}, B = – 4x^{2} + xy + 5y^{2} and C = 4y^{2} â€“ 3x^{2} â€“ 6xy then show that A + B + C = 0.**

**Solution**

Given

A = 7x^{2} + 5xy â€“ 9y^{2}

B = – 4x^{2} + xy + 5y^{2}

C = 4y^{2} â€“ 3x^{2} â€“ 6xy

To show A + B + C = 0

Substitute the value of A, B and C in A + B + C

A + B + C = (7x^{2} + 5xy â€“ 9y^{2}) + (- 4x^{2} + xy + 5y^{2}) + (4y^{2} â€“ 3x^{2} â€“ 6xy)

= 7x^{2} + 5xy â€“ 9y^{2} â€“ 4x^{2} + xy + 5y^{2} + 4y^{2} â€“ 3x^{2} â€“ 6xy

Rearranging and collecting the like terms, we get:

= 7x^{2} â€“ 4x^{2} â€“ 3x^{2} + 5xy + xy â€“ 6xy â€“ 9y^{2 }+ 5y^{2} + 4y^{2}

= (7 â€“ 4 â€“ 3) x^{2} + (5 + 1 â€“ 6) xy â€“ (9 â€“ 5 â€“ 4) y^{2}

= 0 x^{2} + 0 xy â€“ 0 y^{2}

^{ } = 0 + 0 + 0

= 0

Hence,

A + B + C = 0

**9. What must be added to 5x ^{3} â€“ 2x^{2} + 6x + 7 to make the sum x^{3} + 3x^{2} â€“ x + 1**

**Solution**

** **Let X be the expression to be added to 5x^{3} â€“ 2x^{2} + 6x + 7

(5x^{3} â€“ 2x^{2} + 6x + 7) + X = x^{3} + 3x^{2} â€“ x + 1

X = (x^{3} + 3x^{2} â€“ x + 1) â€“ (5x^{3} â€“ 2x^{2} + 6x + 7)

Changing the sign of each term of expression to be subtracted and add it to the expression from which subtraction is to be made

Term which is subtracted = 5x^{3} â€“ 2x^{2} + 6x + 7

Changing the sign of each term of expression = – 5x^{3 }+ 2x^{2} â€“ 6x â€“ 7

X = (x^{3} + 3x^{2} â€“ x + 1) + (- 5x^{3} + 2x^{2} â€“ 6x â€“ 7)

= x^{3} + 3x^{2} â€“ x + 1 – 5x^{3} + 2x^{2} – 6x – 7

Rearranging and collecting the like terms, we get:

= x^{3} – 5x^{3} + 3x^{2} + 2x^{2} â€“ x – 6x + 1 – 7

= (1 – 5) x^{3 }+ (3 + 2) x^{2} â€“ (1 + 6) x + 1 – 7

= – 4 x^{3} + 5 x^{2} + 7 x â€“ 6

âˆ´ – 4 x^{3} + 5 x^{2} + 7 x â€“ 6 must be added to 5x^{3} â€“ 2x^{2} + 6x + 7 to make the sum x^{3} + 3x^{2} â€“ x + 1

## Exercise 8D PAGE NO: 136

**Simplify:**

**1. a â€“ (b -2a)**

**Solution**

** **Here, (-) sign precedes the second parenthesis, so we remove it and change the sign of each term within

âˆ´ a â€“ (b â€“ 2a)

= a â€“ (b â€“ 2a)

= a â€“ b + 2a

= 3a â€“ b

**2. 4x â€“ (3y â€“ x + 2z)**

**Solution**

** **Here, (-) sign precedes the second parenthesis, so we remove it and change the sign of each term within

** **âˆ´ 4x â€“ (3y â€“ x + 2z)

= 4x â€“ 3y + x â€“ 2z

= 4x + x â€“ 3y â€“ 2z

= 5x â€“ 3y â€“ 2z

**3. (a ^{2} + b^{2} + 2ab) â€“ (a^{2} + b^{2} -2ab)**

**Solution**

** **Here, (-) sign precedes the second parenthesis, so we remove it and change the sign of each term within

** **âˆ´ (a^{2} + b^{2} + 2ab) â€“ (a^{2} + b^{2} -2ab)

= a^{2} + b^{2} + 2ab â€“ a^{2 }â€“ b^{2 }+ 2ab

= a^{2} â€“ a^{2} + b^{2 }â€“ b^{2} + 2ab + 2ab

= 4ab

**4. – 3(a + b) + 4 (2a â€“ 3b) â€“ (2a â€“ b)**

**Solution**

Here, (-) sign precedes the first and third parenthesis, so we remove it and change the sign of each term within

** **âˆ´ – 3(a + b) + 4 (2a â€“ 3b) â€“ (2a â€“ b)

= -3a â€“ 3b + 4 (2a) â€“ 4 (3b) â€“ 2a + b

= -3a â€“ 3b + 8a â€“ 12b â€“ 2a + b

= – 3a + 8a â€“ 2a â€“ 3b â€“ 12b + b

= 8a â€“ 5a â€“ 15b + b

= 3a -14b

**5. -4x ^{2} + {(2x^{2} â€“ 3) â€“ (4 â€“ 3x^{2})}**

**Solution**

We first remove the innermost grouping symbol ( ) and then { },

We have:

**–**4x^{2} + {(2x^{2} â€“ 3) â€“ (4 â€“ 3x^{2})}

= – 4x^{2 }+ {2x^{2} â€“ 3 â€“ 4 + 3x^{2}}

= – 4x^{2} + 2x^{2} â€“ 3 â€“ 4 + 3x^{2}

= – 4x^{2 }**+ **2x^{2} + 3x^{2} â€“ 3 â€“ 4

= – 4x^{2} + 5x^{2} â€“ 7

= x^{2 }â€“ 7

**6. – 2(x ^{2} â€“ y^{2} + xy) â€“ 3(x^{2} **+

**y**

^{2}– xy)**Solution**

** **Here, (-) sign precedes both the parenthesis, so we remove it and change the sign of each term within

âˆ´ – 2(x^{2} â€“ y^{2} + xy) â€“ 3(x^{2} + y^{2} – xy)

= -2x^{2} + 2y^{2} â€“ 2xy â€“ 3x^{2 }**– **3y^{2} + 3xy

= – 2x^{2} -3x^{2} + 2y^{2} â€“ 3y^{2} â€“ 2xy + 3xy

= (-2 â€“ 3) x^{2} + (2 â€“ 3) y^{2} + (-2 + 3) xy

= – 5x^{2} â€“ y^{2} + xy

**7. a â€“ [2b â€“ {3a – (2b -3c)}]**

**Solution**

We first remove the innermost grouping symbol ( ), { } and [ ]

We have,

a â€“ [2b â€“ {3a – (2b -3c)}]

= a â€“ [2b â€“ {3a â€“ 2b + 3c}]

= a â€“ [2b â€“ 3a + 2b â€“ 3c]

= a â€“ 2b + 3a â€“ 2b + 3c

= a â€“ 3a â€“ 2b â€“ 2b + 3c

= 4a â€“ 4b + 3c

**8. – x + [5y â€“ {x â€“ (5y â€“ 2x)}]**

**Solution**

We first remove the innermost grouping symbol ( ), { } and [ ]

We have

** **– x + [5y â€“ {x â€“ (5y â€“ 2x)}]

= – x + [5y â€“ {x â€“ 5y + 2x}]

= – x + [5y â€“ x + 5y â€“ 2x]

= – x + 5y â€“ x + 5y â€“ 2x

= – x â€“ x â€“ 2x + 5y + 5y

= (-1 -1 -2) x + (5 + 5) y

= – 4x + 10y

**9. 86 â€“ [15x â€“ 7(6x â€“ 9) â€“ 2{10x â€“ 5(2 â€“ 3x)}]**

**Solution**

** **We first remove the innermost grouping symbol ( ), { } and [ ]

** **We have

86 â€“ [15x â€“ 7(6x â€“ 9) â€“ 2{10x â€“ 5(2 â€“ 3x)}]

= 86 â€“ [15x â€“ 42x + 63 â€“ 2{10x â€“ 10 + 15x}]

= 86 â€“ [15x â€“ 42x + 63 â€“ 20x + 20 â€“ 30x]

= 86 â€“ 15x + 42x â€“ 63 + 20x â€“ 20 + 30x

= 86 â€“ 63 â€“ 20 â€“ 15x + 42x + 20x + 30x

= 3 + 77x

### RS Aggarwal Solutions for Class 6 Chapter 8 Algebraic Expressions

Chapter 8 – Algebraic Expressions consists of 4 exercises. RS Aggarwal Solutions have been solved in detail for each question in every exercise. Letâ€™s have a glance at the topics included in this chapter

- Operations on Literals and Numbers
- Algebraic Expressions
- Operations on Algebraic Expressions
- Use of grouping symbols

Also, access RS Aggarwal Solutions for Class 6 Chapter 8 Exercises

### Chapter Brief of RS Aggarwal Solutions for Class 6 Maths Chapter 8 – Algebraic Expressions

An expression having both variable and constant along with algebraic operations such as addition, subtraction, multiplication and division. For example, 6x – 4,

here x is the variable whose value is unknown, 6 is known as coefficient of x and 4 is the constant term having a definite value. There are five types of algebraic expressions.

(i) Monomials – expression having one term known as monomials

(ii) Binomials – expression having two terms known as binomials

(iii) Trinomials – expression having three terms known as trinomials

(iv) Quadrinomials – expression having four terms known as quadrinomials

(v) Polynomials – expression having two or more terms known as polynomials

Algebraic Expressions are used in cooking, business management, sports, logical thinking.