RS Aggarwal Solutions Class 6 Ex 8A

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

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

=(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\(a^{3}\), – 4\(a^{3}\), 10\(a^{3}\) , -8\(a^{3}\)

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

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

= 4\(a^{3}\)

(viii) \(x^{2}\)\(a^{2}\), -5\(x^{2}\) + 2\(a^{2}\)), – 4\(x^{2}\) + 4\(a^{2}\)

Required sum = \(x^{2}\)\(a^{2}\) + (-5\(x^{2}\) + 2\(a^{2}\)) +( – 4\(x^{2}\) + 4\(a^{2}\) )

Rearranging and collecting the like terms = \(x^{2}\) -5\(x^{2}\) – 4\(x^{2}\)\(a^{2}\) + 2\(a^{2}\) + 4\(a^{2}\)

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

= -8\(x^{2}\) + 5\(a^{2}\)

2)

(i)

x- 3y -2z

5 x + 7y – z

-7x – 2y + 4z

-x +2y + z

(ii)

\(m^{2}\) – 4m + 5 – 2\(m^{2}\) + 6m – 6 – \(m^{2}\) – 2m – 7 -2\(m^{2}\) +0x m-8 = -2\(m^{2}\) – 8 = -2\(m^{2}\) – 8

(iii)

2 x 2 – 3xy + \(y^{2}\) – 7\(x^{2}\) – 5xy – 2\(y^{2}\) 4\(x^{2}\) + xy – 6\(y^{2}\)\(x^{2}\) -7xy -7\(y^{2}\)

(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\(x^{3}\) – 3\(x^{2}\) + 7x – 8,-5\(x^{3}\) + 2\(x^{2}\) – 4x + 1, 3 – 6x + 5\(x^{2}\)\(x^{3}\)

Sum of the given expressions

= (2\(x^{3}\) – 3\(x^{2}\) + 7x – 8) + (-5\(x^{3}\) + 2\(x^{2}\) – 4x + 1) + ( 3 – 6x + 5×2 – x3) Rearranging and collecting the like terms =2×3-5×3 – x3 – 3×2 + 2×2 + 5×2 +7x-4x-6x-8+1+3 = (2-5-1)x3 +(-3+2+5)x2+(7-4-G)x-4 = -4×3 +4×2-3x-4

(iv) 2\(x^{2}\) – 8xy + 7\(y^{2}\) – 8x\(y^{2}\), 2x\(y^{2}\) + 6xy – \(y^{2}\) + 3\(x^{2}\), 4\(y^{2}\) – xy – \(x^{2}\) + x\(y^{2}\)

Sum of the given expressions = (2\(x^{2}\) – 8xy + 7\(y^{2}\) – 8x\(y^{2}\))+( 2x\(y^{2}\) + 6xy – \(y^{2}\) + 3\(x^{2}\)) + ( 4\(y^{2}\) – xy – \(x^{2}\) + x\(y^{2}\) )

Rearranging and collecting the like terms

= 2\(x^{2}\) +3\(x^{2}\)\(x^{2}\) + 7\(y^{2}\)\(y^{2}\) +4\(y^{2}\) – 8xy + 6xy – xy – 8x\(y^{2}\) +2x\(y^{2}\) + x\(y^{2}\)

= (2 +3- 1) \(x^{2}\) + (7 – 1 +4) \(y^{2}\) + (-8 + 6 -1)xy + (- 8 +2 +1)x\(y^{2}\)

= 4\(x^{2}\) + 10\(y^{2}\) – 3xy -5x\(y^{2}\)

(v) \(x^{3}\) + \(y^{3}\)\(z^{3}\) + 3xyz, – \(x^{3}\) + \(y^{3}\) + \(z^{3}\) – 6xyz, \(x^{3}\)\(y^{3}\)\(z^{3}\) – 8xyz

Sum of the given expressions = (\(x^{3}\) + \(y^{3}\)\(z^{3}\) + 3xyz)+(- \(x^{3}\) + \(y^{3}\) + \(z^{3}\) – 6xyz)+( \(x^{3}\)\(y^{3}\)\(z^{3}\) – 8xyz)

Rearranging and collecting the like terms

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

= (1-1+1) \(x^{3}\) + (1+1-1) \(y^{3}\) + (-1+1-1) \(z^{3}\) +(3-6-8)xyz

= \(x^{3}\) + ya – \(z^{3}\) -11xyz

(vi) 2 + x – \(x^{2}\) + 6\(x^{3}\), -6 -2x + 4\(x^{2}\) -3\(x^{3}\), 2 + \(x^{2}\), 3 – \(x^{3}\) + 4x – 2\(x^{2}\)

Sum of the given expressions

= (2 + x – \(x^{2}\) + 6\(x^{3}\)) + (-6 -2x + 4\(x^{2}\) -3\(x^{3}\))+( 2 + \(x^{2}\))+( 3 – \(x^{3}\) + 4x – 2\(x^{2}\) )

Rearranging and collecting the like terms

= 6\(x^{3}\) – 3\(x^{3}\)\(x^{3}\)\(x^{2}\) + 4\(x^{2}\) + \(x^{2}\) – 2\(x^{2}\) + x – 2x+ 4x+2 – 6+2+3

= (6-3-1) \(x^{3}\) + (-1+4+1-2) \(x^{2}\) +(1-2+4)x + 1

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

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\(x^{2}\) from -7\(x^{2}\)

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

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

On adding:

-7\(x^{2}\) + (-10\(x^{2}\)) = -7\(x^{2}\) -10\(x^{2}\) = (-7-10) \(x^{2}\) = -17\(x^{2}\)

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

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

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

On adding:

\(b^{2}\)\(a^{2}\) + (-\(a^{2}\) + \(b^{2}\)) = \(b^{2}\)\(a^{2}\)\(a^{2}\) + \(b^{2}\)

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

= 2\(b^{2}\) – 2\(a^{2}\)

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\(x^{2}\) – 3xy + \(y^{2}\) from 7\(x^{2}\) – 2xy – 4\(y^{2}\)

Term to be subtracted = 5\(x^{2}\) – 3xy + \(y^{2}\)

Changing the sign of each term of the expression gives -5\(x^{2}\) + 3xy – \(y^{2}\).

On adding:

(7\(x^{2}\) – 2xy – 4\(y^{2}\))+(-5\(x^{2}\) + 3xy – \(y^{2}\))

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

= (7-5) \(x^{2}\) + (-2+3)xy +(-4-1) \(y^{2}\)

= 2\(x^{2}\) + xy – 5\(y^{2}\)

(iv) 6\(x^{3}\) – 7\(x^{2}\) + 5x – 3 from 4 – 5x + 6\(x^{2}\) – 8\(x^{3}\)

Term to be subtracted = 6\(x^{3}\) – 7\(x^{2}\) + 5x – 3

Changing the sign of each term of the expression gives -6\(x^{3}\) + 7\(x^{2}\) – 5x + 3.

On adding:

(4 – 5x + 6\(x^{2}\) – 8\(x^{3}\))+(-6\(x^{3}\) + 7\(x^{2}\) – 5x + 3)

= 4 – 5x + 6\(x^{2}\) – 8\(x^{3}\) -6\(x^{3}\) + 7\(x^{2}\) – 5x + 3

= (-8-6) \(x^{3}\) +(6+7) \(x^{2}\) +(-5- 5)x + 7

= -14\(x^{3}\) + 13\(x^{2}\) – 10x + 7

(v) \(x^{3}\) + 2\(x^{2}\) y + 6x\(y^{2}\)\(y^{3}\) from (\(y^{3}\) – 3x\(y^{2}\) – 4\(x^{2}\)y

Term to be subtracted = \(x^{3}\) + 2\(x^{2}\) y + 6x\(y^{2}\)\(y^{3}\)

Changing the sign of each term of the expression gives

\(x^{3}\) – 2\(x^{2}\)y – 6x\(y^{2}\) + \(y^{3}\).

On adding: (\(y^{3}\) – 3x\(y^{2}\) – 4\(x^{2}\)y) + (-\(x^{3}\) – 2\(x^{2}\) y – 6x\(y^{2}\) + \(y^{3}\))

= \(y^{3}\) – 3x\(y^{2}\) – 4\(x^{2}\) y – \(x^{3}\) – 2\(x^{2}\) y – 6x\(y^{2}\) + \(y^{3}\)

= – \(x^{3}\) +(- 2-4) \(x^{2}\) y + (-6-3)x\(y^{2}\) + (1+1) \(y^{3}\)

= –\(x^{3}\) – 6\(x^{2}\) y – 9x\(y^{2}\) + 2\(y^{3}\)

(vi) -11\(x^{2}\) \(y^{2}\) + 7xy – 6 from 9\(x^{2}\) \(y^{2}\) – 6xy + 9

Term to be subtracted = -11\(x^{2}\) \(y^{2}\) + 7xy – 6

Changing the sign of each term of the expression gives 11\(x^{2}\) \(y^{2}\) – 7xy + 6.

On adding:

(9\(x^{2}\) \(y^{2}\) – 6xy + 9) + (11\(x^{2}\) \(y^{2}\) – 7xy +6)

= 9\(x^{2}\) \(y^{2}\) – 6xy + 9 +11\(x^{2}\) \(y^{2}\) -7xy +6

= (9+11) \(x^{2}\) \(y^{2}\) (-7-6)xy + 15

= 20\(x^{2}\) \(y^{2}\) -13xy +15

(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\(p^{3}\) – 3\(p^{2}\) + 4p – 5 – 6\(p^{3}\) + 2\(p^{2}\) – 8p – 2 + 6p + 8

Rearranging and collecting the like terms

= (2-6) \(p^{3}\) +(-3+2) \(p^{2}\) + (4-8+6)p – 5 – 2 + 8

= -4\(p^{3}\)\(p^{2}\) +2p +1

(ii) 2\(x^{2}\) – xy + 6x – 4y + 5xy – 4x + 6\(x^{2}\) + 3y

Rearranging and collecting the like terms = (2+6) \(x^{2}\) +(-1+5) xy + (6-4)x +(- 4+3)y

= 8\(x^{2}\) + 4xy + 2x – y

(iii) \(x^{4}\) – 6\(x^{3}\) + 2x – 7 + 7×3 – x + 5\(x^{2}\) + 2 – \(x^{4}\)

Rearranging and collecting the like terms

= (1-1) \(x^{4}\) +(- 6+7) \(x^{3}\) + 5\(x^{2}\) +(2-1)x -7+ 2

= 0 + \(x^{3}\) + 5\(x^{2}\) + x – 5

= \(x^{3}\) + 5\(x^{2}\) + x – 5.

7) From the sum of 3\(x^{2}\) – 5x + 2 and -5\(x^{2}\)– 8x + 6, subtract 4\(x^{2}\) – 9x + 7

Adding:

(3\(x^{2}\) – 5x + 2) + (-5\(x^{2}\)– 8x + 6)

Rearranging and collecting the like terms:

(3-5) \(x^{2}\) +(- 5-8)x + 2 +6

= -2\(x^{2}\) – 13x + 8

Subtract 4\(x^{2}\) – 9x + 7 from -2\(x^{2}\) – 13x +

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

Term to be subtracted = 4\(x^{2}\) – 9x + 7

Changing the sign of each term of the expression gives -4\(x^{2}\) + 9x – 7.

On adding:

( -2\(x^{2}\) – 13x + 8 )+(-4\(x^{2}\) + 9x – 7 ) = -2\(x^{2}\)– 13x + 8 -4\(x^{23}\) + 9x – 7

= ( -2-4) \(x^{2}\) + (-13+9)x + 8 -7

= -6\(x^{2}\) – 4x + 1

8) If A = 7\(x^{2}\) + 5xy -9\(y^{2}\) , B = -4\(x^{2}\) + xy + 5\(y^{2}\) and C = 4\(y^{2}\) – 3\(x^{2}\) – 6xy then show that A + B + C = 0.

A = 7\(x^{2}\) + 5xy – 9\(y^{2}\)

B = -4\(x^{2}\) + xy + 5\(y^{2}\)

C = 4y2 – 3×2 – 6xy

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

= (7\(x^{2}\) + 5xy – 9\(y^{2}\)) + (-4\(x^{2}\) + xy + 5\(y^{2}\)) + (4\(y^{2}\) – 3\(x^{2}\) – 6xy)

= 7\(x^{2}\) + 5xy – 9\(y^{2}\) – 4\(x^{2}\) + xy + 5\(y^{2}\) + 4\(y^{2}\) – 3\(x^{2}\) – 6xy

Rearranging and collecting the like terms:

(7-4-3) \(x^{2}\) + (6+1-6)xy +(-9+5+4) \(y^{2}\)

= (0)\(x^{2}\) + (0)xy + (0) \(y^{2}\)

=0

=>A + B + C = 0

9) What must be added to 5\(x^{3}\) – 2 \(x^{2}\) + 6x + 7 to make the sum \(x^{3}\) + 3\(x^{2}\) – 6xy then show that A + B +C = 0.

Let the expression to be added be X.

(5\(x^{3}\) – 2\(x^{2}\) + 6x + 7)+X = (\(x^{3}\) + 3\(x^{2}\) – x + 1)

X= (\(x^{3}\) + 3\(x^{2}\) – x + 1) – (5\(x^{3}\) – 2\(x^{2}\) + 6x + 7)

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

X= (\(x^{3}\) + 3\(x^{2}\) – x + 1) + (-5\(x^{3}\) + 2\(x^{2}\) – 6x – 7)

X = \(x^{3}\) + 3\(x^{2}\) – x + 1 – 5\(x^{3}\) + 2\(x^{2}\) – 6x – 7

Rearranging and collecting the like terms:

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

X = -4\(x^{3}\) + 5\(x^{2}\) – 7x-6

So, -4\(x^{3}\) + 5\(x^{2}\) – 7x -6 must be added to 5\(x^{3}\) – 2\(x^{2}\) + 6x + 7 to get the sum as \(x^{3}\) + 3\(x^{2}\) – x + 1.

10) Let P = \(a^{2}\)\(b^{2}\) + 2ab, Q = \(a^{2}\) + 4\(b^{2}\) – 6ab, R = \(b^{2}\) +6, S =\(a^{2}\) – 4ab T = -2\(a^{2}\) + \(b^{2}\) – ab + a. Find P + Q + R + S – T.

P = \(a^{2}\)\(b^{2}\) + 2ab

Q = \(a^{2}\) + 4\(b^{2}\) – 6ab

R= \(b^{2}\) +6

S = \(a^{2}\) – 4ab

T = -2\(a^{2}\) + \(b^{2}\) – ab + a

Adding P, 0, R and 5: P+Q+R+S = (\(a^{2}\)\(b^{2}\) + 2ab)+( \(a^{2}\) + 4\(b^{2}\) – 6ab)+( \(b^{2}\) + 6)+(\(a^{2}\) – 4ab )

= \(a^{2}\)\(b^{2}\) + 2ab + \(a^{2}\) + 4\(b^{2}\) – 6ab+\(b^{2}\) + 6+\(a^{2}\) – 4ab

Rearranging and collecting the like terms:

= (1+1+1) \(a^{2}\) +(-1+4+1) \(b^{2}\) + (2-6-4)ab+6

P++R+S = 3\(a^{2}\)+ 4\(b^{2}\) – 8ab + 6

To find P + Q + R + S – T, subtract T = (-2\(a^{2}\) + \(b^{2}\) – ab + a) from P+Q+R+S = (3\(a^{2}\) +4\(b^{2}\) – 8ab+6).

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

Term to be subtracted = -2\(a^{2}\) + \(b^{2}\) – ab + a

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

Now add:

(3\(a^{2}\) +4\(b^{2}\) – 8ab+6)+(2\(a^{2}\)\(b^{2}\) + ab – a) = 3\(a^{2}\) +4\(b^{2}\) – 8ab + 6 + 2\(a^{2}\)\(b^{2}\) + ab – a

= (3+2) \(a^{2}\) + (4-1) \(b^{2}\) +(-8+1) ab – a+6

P + Q + R + S – T = 5\(a^{2}\) +3\(b^{2}\) -7ab – a+6

11) What must be subtracted from \(a^{3}\) – 4\(a^{2}\) + 5a – 6 to obtain \(a^{2}\) – 2a + 1?

Let the expression to be subtracted be X.

(\(a^{3}\) – 482 + 5a – 6)-X = (\(a^{2}\) – 2a + 1)

X = (\(a^{3}\) – 4\(a^{2}\) + 5a – 6)- (\(a^{2}\) – 2a + 1)

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

X = \(a^{3}\) – 4\(a^{2}\) + 5a – 6 – \(a^{2}\) + 2a – 1

Rearranging and collecting the like terms:

X = \(a^{3}\) +(- 4-1)\(a^{2}\) + (5+2)a – 6 – 1

X = \(a^{3}\) -5\(a^{2}\) + 7a – 7

So, \(a^{3}\) – 5\(a^{2}\) + 7a – 7 must be subtracted from \(a^{3}\) – 4\(a^{2}\) + 5a – 6 to obtain \(a^{2}\) – 2a + 1.

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\(x^{2}\) – 5x + 6 exceed \(x^{3}\)\(x^{2}\) + 4x – 1 ?

To calculate how much does 3\(x^{2}\) – 5x + 6 exceed \(x^{3}\)\(x^{2}\) + 4x – 1, we have to subtract \(x^{3}\)\(x^{2}\) + 4x – 1 from 3\(x^{2}\) – 5x + 6.

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

Term to be subtracted = \(x^{3}\)\(x^{2}\) + 4x – 1

Changing the sign of each term of the expression gives –\(x^{3}\) + \(x^{2}\) – 4x + 1.

On adding:

(3\(x^{2}\) – 5x + 6)+(- \(x^{3}\) + \(x^{2}\) – 4x + 1 )

= 3\(x^{2}\) – 5x + 6 – \(x^{3}\) + \(x^{2}\) – 4x + 1

= –\(x^{3}\) + (3+1) \(x^{2}\) +(-5-4)x + 6 + 1

= –\(x^{3}\) +4 \(x^{2}\) – 9x + 7

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


Practise This Question

Given below is a pattern made using squares. Observe the pattern and tell the number of squares in the next figure in the sequence. Also find the general form which represents the number of squares in the sequence.