1) Add:

(i) 3x, 7x

Required sum = 3x + 7x

= (3+7)x

= 10x

(ii) 7y, -9y

Required sum = 7y +(-9y)

= (7-9)y

= -2y

(iii) 2xy, 5xy, -xy

Required sum = 2xy +5xy + (-xy)

= (2+5-1)xy

= 6xy

(iv) 3x, 2y

Required sum = 3x+2y

(v) 2\(x^{2}\)

Required sum = 2\(x^{2}\)

=(2-3+7) \(x^{2}\)

= 6\(x^{2}\)

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

Required sum = 7xyz + (- 5xyz) + 9xyz + (-8xyz)

= (7-5+9-8) xyz

= 3xyz

(vii) 6\(a^{3}\)

Required sum = 6\(a^{3}\)

=(6-4+10-8) \(a^{3}\)

= 4\(a^{3}\)

(viii) \(x^{2}\)

Required sum = \(x^{2}\)

Rearranging and collecting the like terms = \(x^{2}\)

= (1-5-4) \(x^{2}\)

= -8\(x^{2}\)

2)

(i)

x- 3y -2z

5 x + 7y – z

-7x – 2y + 4z

-x +2y + z

(ii)

\(m^{2}\)

(iii)

2 x 2 – 3xy + \(y^{2}\)

(iv)

4xy – 5yz – 7zx- 5xy + 2yz + zx- 2xy – 3yz + 3zx-3xy -6yz -3zx

3) Add:

(i) (3a – 2b + 5c), (2a + 5b – 7c), (- a – b + c)

Sum of the given expressions

= (3a – 2b + 5c) + (2a + 5b – 7c) + (- a – b + c)

Rearranging and collecting the like terms

= 3a + 2a – a – 2b + 5b – b + 5c – 7c + c

= (3+2-1)a + (-2+5-1)b + (5-7+1)c

= 4a+2b-c

(ii) Sum of the given expressions

(8a – 6ab + 5b), (- 6a – ab – 8b), (-4a + 2ab + 3b)

= (8a – 6ab + 5b) + (- 6a – ab – 8b) + (-4a + 2ab + 3b)

Rearranging and collecting the like terms

= (8 – 6 – 4 )a + (- 6 – 1 + 2)ab + (5 – 8 + 3)b

= -2a – 5ab + 0

= -2a – 5ab

(iii) 2\(x^{3}\)

Sum of the given expressions

= (2\(x^{3}\)

(iv) 2\(x^{2}\)

Sum of the given expressions = (2\(x^{2}\)

Rearranging and collecting the like terms

= 2\(x^{2}\)

= (2 +3- 1) \(x^{2}\)

= 4\(x^{2}\)

(v) \(x^{3}\)

Sum of the given expressions = (\(x^{3}\)

Rearranging and collecting the like terms

= 6\(x^{3}\)

= (1-1+1) \(x^{3}\)

= \(x^{3}\)

(vi) 2 + x – \(x^{2}\)

Sum of the given expressions

= (2 + x – \(x^{2}\)

Rearranging and collecting the like terms

= 6\(x^{3}\)

= (6-3-1) \(x^{3}\)

= 2\(x^{3}\)

4) Subtract:

Change the sign of each term of the expression that is to be subtracted and then add.

(i) 5x from 2x

Term to be subtracted = 5x

Changing the sign of each term of the expression gives -5x.

On adding:

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

= (2-5)x

= -3x

(ii) –xy from 6xy

Term to be subtracted = -xy

Changing the sign of each term of the expression gives xy.

On adding:

6xy + xy

= (6+1)xy

= 7xy

(iii) 3a from 5b

Term to be subtracted = 3a

Changing the sign of each term of the expression gives -3a.

On adding:

5b+(-3a) = 5b-3a

(iv) -7x from 9y

Term to be subtracted = -7x

Changing the sign of each term of the expression gives 7x.

On adding: 9y+7x

(v) 10\(x^{2}\)

Term to be subtracted = 10\(x^{2}\)

Changing the sign of each term of the expression gives -10\(x^{2}\)

On adding:

-7\(x^{2}\)

(vi) \(a^{2}\)

Term to be subtracted = \(a^{2}\)

Changing the sign of each term of the expression gives –\(a^{2}\)

On adding:

\(b^{2}\)

= (1+1) \(b^{2}\)

= 2\(b^{2}\)

5) Subtract:

Change the sign of each term of the expression that is to be subtracted and then add.

(i) 5a + 7b – 2c from 3a – 7b + 4c

Term to be subtracted = 5a + 7b – 2c

Changing the sign of each term of the expression gives -5a -7b + 2c.

On adding:

(3a – 7b + 4c) + (-5a -7b + 2c ) = 3a – 7b + 4c-5a -7b + 2c

= (3-5)a+( – 7-7)b + (4+2)c

= -2a – 14b + 6c

(ii) a – 2b – 3c from -2a + 5b – 4c

Term to be subtracted = a – 2b – 3c

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

On adding:

(-2a + 5b – 4c)+(-a +2b + 3c ) = -2a + 5b – 4c-a +2b + 3c

= (-2-1)a + (5+2)b +(-4+3)c

= -3a + 7b – c

(iii) 5\(x^{2}\)

Term to be subtracted = 5\(x^{2}\)

Changing the sign of each term of the expression gives -5\(x^{2}\)

On adding:

(7\(x^{2}\)

= 7\(x^{2}\)

= (7-5) \(x^{2}\)

= 2\(x^{2}\)

(iv) 6\(x^{3}\)

Term to be subtracted = 6\(x^{3}\)

Changing the sign of each term of the expression gives -6\(x^{3}\)

On adding:

(4 – 5x + 6\(x^{2}\)

= 4 – 5x + 6\(x^{2}\)

= (-8-6) \(x^{3}\)

= -14\(x^{3}\)

(v) \(x^{3}\)

Term to be subtracted = \(x^{3}\)

Changing the sign of each term of the expression gives

–\(x^{3}\)

On adding: (\(y^{3}\)

= \(y^{3}\)

= – \(x^{3}\)

= –\(x^{3}\)

(vi) -11\(x^{2}\)

Term to be subtracted = -11\(x^{2}\)

Changing the sign of each term of the expression gives 11\(x^{2}\)

On adding:

(9\(x^{2}\)

= 9\(x^{2}\)

= (9+11) \(x^{2}\)

= 20\(x^{2}\)

(vii) -2a + b + 6d from 5a – 2b -3c

Term to be subtracted = -2a + b + 6d

Changing the sign of each term of the expression gives 2a-b-6d.

On adding:

(5a – 2b -3c) + (2a-b-6d ) = 5a – 2b -3c +2a-b-6d

= (5+2)a+(- 2-1)b -3c -6d

= 7a – 3b-3c -6d

6) Simplify:

(i) 2\(p^{3}\)

Rearranging and collecting the like terms

= (2-6) \(p^{3}\)

= -4\(p^{3}\)

(ii) 2\(x^{2}\)

Rearranging and collecting the like terms = (2+6) \(x^{2}\)

= 8\(x^{2}\)

(iii) \(x^{4}\)

Rearranging and collecting the like terms

= (1-1) \(x^{4}\)

= 0 + \(x^{3}\)

= \(x^{3}\)

7) From the sum of 3\(x^{2}\)

Adding:

(3\(x^{2}\)

Rearranging and collecting the like terms:

(3-5) \(x^{2}\)

= -2\(x^{2}\)

Subtract 4\(x^{2}\)

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = 4\(x^{2}\)

Changing the sign of each term of the expression gives -4\(x^{2}\)

On adding:

( -2\(x^{2}\)

= ( -2-4) \(x^{2}\)

= -6\(x^{2}\)

8) If A = 7\(x^{2}\)

A = 7\(x^{2}\)

B = -4\(x^{2}\)

C = 4y2 – 3×2 – 6xy

Substituting the values of A, B and C in A+B+C:

= (7\(x^{2}\)

= 7\(x^{2}\)

Rearranging and collecting the like terms:

(7-4-3) \(x^{2}\)

= (0)\(x^{2}\)

=0

=>A + B + C = 0

9) What must be added to 5\(x^{3}\)

Let the expression to be added be X.

(5\(x^{3}\)

X= (\(x^{3}\)

Changing the sign of each term of the expression that is to be subtracted and then adding:

X= (\(x^{3}\)

X = \(x^{3}\)

Rearranging and collecting the like terms:

X = (1-5) \(x^{3}\)

X = -4\(x^{3}\)

So, -4\(x^{3}\)

10) Let P = \(a^{2}\)

P = \(a^{2}\)

Q = \(a^{2}\)

R= \(b^{2}\)

S = \(a^{2}\)

T = -2\(a^{2}\)

Adding P, 0, R and 5: P+Q+R+S = (\(a^{2}\)

= \(a^{2}\)

Rearranging and collecting the like terms:

= (1+1+1) \(a^{2}\)

P++R+S = 3\(a^{2}\)

To find P + Q + R + S – T, subtract T = (-2\(a^{2}\)

On changing the sign of each term of the expression that is to be subtracted and then adding:

Term to be subtracted = -2\(a^{2}\)

Changing the sign of each term of the expression gives 2\(a^{2}\)

Now add:

(3\(a^{2}\)

= (3+2) \(a^{2}\)

P + Q + R + S – T = 5\(a^{2}\)

11) What must be subtracted from \(a^{3}\)

Let the expression to be subtracted be X.

(\(a^{3}\)

X = (\(a^{3}\)

Since ‘-‘ sign precedes the parenthesis, we remove it and change the sign of each term within the parenthesis.

X = \(a^{3}\)

Rearranging and collecting the like terms:

X = \(a^{3}\)

X = \(a^{3}\)

So, \(a^{3}\)

12) How much is a + 2b – 3c greater than 2a – 3b + c ?

To calculate how much is a + 2b – 3c greater than 2a – 3b + c, we have to subtract 2a – 3b + c from a + 2b – 3c.

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = 2a – 3b + c

Changing the sign of each term of the expression gives -2a + 3b – c.

On adding:

(a + 2b – 3c )+(-2a + 3b – c )

= a + 2b – 3c -2a + 3b – c

= (1-2)a + (2+3)b +(- 3-1)c

= – a + 5b – 4c

13) How much less than x – 2y + 3z is 2x – 4y – z ?

To calculate how much less than x – 2y + 3z is 2x – 4y – z, we have to subtract 2x – 4y – z from x – 2y + 3z.

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = 2x – 4y – z

Changing the sign of each term of the expression gives -2x + 4y + z.

On adding:

(x – 2y + 3z)+(-2x + 4y + z )

= x – 2y + 3z-2x + 4y + z

= (1-2)x +(-2+4)y + (3+1)z

= -X + 2y + 4z

14) By how much does 3\(x^{2}\)

To calculate how much does 3\(x^{2}\)

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = \(x^{3}\)

Changing the sign of each term of the expression gives –\(x^{3}\)

On adding:

(3\(x^{2}\)

= 3\(x^{2}\)

= –\(x^{3}\)

= –\(x^{3}\)

15) Subtract the sum of 5x – 4y + 6z and -8x + y – 2z from the sum of 12x – y + 3z and -3x + 5y – 8z.

Add 5x – 4y + 6z and -8x + y – 2z.

(5x – 4y + 6z ) + (-8x + y – 2z)

= 5x – 4y + 6z -8x + y – 2z

= (5-8)x +(-4+1)y + (6-2)z

= -3x – 3y + 4z

Adding 12x – y + 3z and -3x + 5y – 8z:

(12x – y + 3z )+(-3x + 5y – 8z)

= 12x – y + 3z -3x + 5y – 8z

= (12-3)x +(-1+5)y + (3-8)z

= 9x +4y -5z

Subtract -3x – 3y + 4z from 9x +4y -5z.

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = -3x – 3y + 4z

Changing the sign of each term of the expression gives 3x + 3y – 4z.

On adding:

(9x +4y -5z)+(3x + 3y – 4z )

= 9x +4y -5z+3x + 3y – 4z

= (9+3)x +(4+3)y + (-5-4)z

= 12x +7y -9z

16) By how much is 2x – 3y + 4z greater than 2x + 5y – 6z + 2 ?

To calculate how much is 2x – 3y + 4z greater than 2x + 5y – 6z + 2, we have to subtract 2x + 5y – 6z + 2 from 2x – 3y + 4z.

Change the sign of each term of the expression that is to be subtracted and then add.

Term to be subtracted = 2x + 5y – 6z + 2

Changing the sign of each term of the expression gives -2x – 5y + 6z – 2. On adding:

(2x – 3y + 4z )+(-2x – 5y + 6z – 2 )

= 2x – 3y + 4z-2x – 5y + 6z – 2

= (2-2)x + (-3-5)y +(4+6)z-2

= 0-8y+10z-2

= -8y+10z-2

17) By how much does 1 exceed 2x – 3y – 4 ?

To calculate how much does 1 exceed 2x – 3y – 4, we have to subtract 2x – 3y – 4 from 1.

Change the sign of each term of the expression to be subtracted and then add.

Term to be subtracted = 2x-3y-4

Changing the sign of each term of the expression gives – 2x + 3y + 4.

On adding:

(1)+(-2x+3y+4 )

= 1 – 2x + 3y + 4

= 5 – 2x – 1 – 3y