RD Sharma solution class 7 chapter 7 Algebraic Expression Exercise 7.2

Exercise 7.2

Page-7.13

Question 1:

Add the following:
(i) 3x and 7x
(ii) −5xy and 9xy

Answer 1:

We have
(i) 3x + 7x = (3 + 7)x = 10x
(ii) -5xy + 9xy = ( -5 + 9)xy = 4xy

Question 2:

Simplify each of the following:
(i) 7x³y + 9yx³
(ii) 12a²b + 3ba²

Answer 2:

Simplifying the given expressions, we have
(i) 7x³y + 9yx³ = (7 + 9)x³y = 16x³y
(ii) 12a²b + 3ba² = (12 + 3)a²b = 15a²b

Question 3:

Add the following:
(i) 7abc, −5abc, 9abc, −8abc
(ii) 2x²y, − 4x²y, 6x²y, −5x²y

Answer 3:

Adding the given terms, we have
(i) 7abc + (- 5abc) + (9 abc) + (- 8abc)
    = 7abc - 5abc + 9abc - 8abc
    = (7 - 5 + 9 - 8)abc
    = (16 - 13)abc
    = 3abc

(ii) 2x²y + (- 4x²y) + 6x²y + (- 5x²y)
    = 2x²y - 4x²y + 6x²y - 5x²y
    = (2 - 4 + 6 - 5)x²y
    = (8 - 9)x²y
    = -x²y

Page-7.14

Question 4:

Add the following expressions:
(i) $x^3-2x^{2}y+3x{y}^2-y^3, 2x^3-5x{y}^2+3x^{2}y-4y^3$
(ii) $a^4-2a^{3}b+3a{b}^3+4a^2{b}^2+3b^4,-2a^4-5a{b}^3+7a^{3}b-6a^2{b}^2+b^4$

Answer 4:

Adding the given expressions, we have
(i) x³- 2x²y + 3xy²- y³+ 2x³- 5xy² + 3x²y- 4y³
     Collecting positive and negative like terms together, we get
     x³+ 2x³- 2x²y + 3x²y + 3xy²- 5xy² - y³ - 4y³
   = 3x³ + x²y - 2xy² - 5y³

(ii) (a⁴- 2a³b + 3ab³ + 4a²b² + 3b⁴) + (-2a⁴- 5ab³ + 7a³b - 6a²b² + b⁴)
      a⁴- 2a³b + 3ab³ + 4a²b² + 3b⁴ - 2a⁴- 5ab³ + 7a³b - 6a²b² + b⁴
      Collecting positive and negative like terms together, we get
     a⁴ - 2a⁴ - 2a³b + 7a³b + 3ab³ - 5ab³ + 4a²b² - 6a²b² + 3b⁴ + b⁴
    = - a⁴ + 5a³b - 2ab³ -  2a²b² + 4b⁴

Question 5:

Add the following expressions:
(i) $8a-6ab+5b, -6a-ab-8b \mathrm{and} -4a+2ab+3b$
(ii) $5x^3+7+6x-5x^2, 2x^2-8-9x, 4x-2x^2+3x^3, 3x^3-9x-x^2 \mathrm{and} x-x^2-x^3-4$

Answer 5:

(i) Required expression = (8a - 6ab + 5b) + (- 6a - ab - 8b) + ( - 4a + 2ab + 3b)
     Collecting positive and negative like terms together, we get
     8a - 6a - 4a - 6ab - ab + 2ab + 5b - 8b + 3b
     = 8a - 10a - 7ab + 2ab + 8b - 8b
     = - 2a - 5ab

(ii) Required expression = (5x³ + 7 + 6x - 5x²) + (2x² - 8 - 9x) + (4x - 2x² + 3x³) + (3x³- 9x - x²) + ( x - x² - x³- 4)
      Collecting positive and negative like terms together, we get
      5x³+ 3x³ + 3x³- x³- 5x² + 2x² - 2x² - x²- x² + 6x - 9x + 4x - 9x + x + 7 - 8 - 4
    = 11x³ - x³ - 7x² + 11x - 18x + 7 - 12
    = 10x³ - 7x² - 7x - 5

Question 6:

Add the following:
(i) $$\begin{gathered} x-3y-2z \\ 5x+7y-8z \\ 3x-2y+5z \end{gathered}$$
(ii) $$\begin{gathered} 4ab-5bc+7ca \\ -3ab+2bc-3ca \\ 5ab-3bc+4ca \end{gathered}$$

Answer 6:

(i)  Required expression = (x - 3y - 2z) + (5x +7y - 8z) +(3x - 2y + 5z)
     Collecting positive and negative like terms together, we get
     x + 5x + 3x - 3y + 7y - 2y - 2z - 8z + 5z
  = 9x - 5y + 7y - 10z + 5z
  = 9x + 2y - 5z

(ii) Required expression = (4ab - 5bc + 7ca) + (- 3ab + 2bc - 3ca ) + (5ab - 3bc + 4ca)
      Collecting positive and negative like terms together, we get
      4ab - 3ab + 5ab - 5bc + 2bc - 3bc + 7ca - 3ca + 4ca
   = 9ab - 3ab - 8bc + 2bc + 11 ca  - 3ca
   = 6ab - 6bc + 8ca

Question 7:

Add 2x² − 3x + 1 to the sum of 3x² − 2x and 3x + 7.

Answer 7:

 Sum of 3x² - 2x and 3x + 7
= (3x² - 2x) + ( 3x +7)
= 3x² - 2x + 3x + 7
= (3x² + x  + 7)
Now, required expression = (2x² - 3x + 1) + (3x² + x  + 7)
                                            = 2x² + 3x² - 3x + x + 1 + 7
                                            = 5x² - 2x + 8

Question 8:

Add x² + 2xy + y² to the sum of x² − 3y² and 2x² − y²+ 9.

Answer 8:

Sum of x² - 3y² and 2x² - y² + 9
= (x² - 3y²) + (2x² - y² + 9)
= x² + 2x² - 3y² - y²+ 9
= 3x² - 4y² + 9

Now, required expression = (x² + 2xy + y²) + (3x² - 4y² + 9)
                                       = x² + 3x² + 2xy + y² - 4y² + 9
                                       = 4x² + 2xy  - 3y² + 9

Question 9:

Add a³ + b³ − 3 to the sum of 2a³ − 3b³ − 3ab + 7 and −a³ + b³ + 3ab −9.

Answer 9:

First, we need to find the sum of 2a³ - 3b³ - 3ab + 7 and - a³ + b³ + 3ab - 9.
= (2a³ - 3b³ - 3ab + 7) + (- a³ + b³ + 3ab - 9)
Collecting positive and negative like terms together, we get
= 2a³ - a³- 3b³ + b³ - 3ab + 3ab + 7 - 9
= a³ - 2b³ - 2

Now, the required expression = (a³ + b³ - 3) + (a³ - 2b³ - 2).
                                  = a³ + a³ + b³- 2b³ - 3 - 2
                                  = 2a³ - b³ - 5

Question 10:

Subtract:
(i) 7a²b from 3a²b
(ii) 4xy from −3xy

Answer 10:

(i) Required expression  =  3a²b - 7a²b
                                        = (3 - 7)a²b
                                        = - 4a²b

(ii) Required expression  = - 3 xy - 4xy
                                         = - 7xy
                                      

Question 11:

Subtract:
(i) −4x from 3y
(ii) −2x from −5y

Answer 11:

(i) Required expression = (3y) - (-4x)
                                       = 3y + 4x

(ii) Required expression = (-5y) - (-2x)
                                        = -5y + 2x

Question 12:

Subtract:
(i) $6x^3-7x^2+5x-3 \mathrm{from} 4-5x+6x^2-8x^3$
(ii) $-x^2-3z \mathrm{from} 5x^2-y+z+7$
(iii) $x^3+2x^{2}y+6x{y}^2-y^3 \mathrm{from} y^3-3x{y}^2-4x^{2}y$

Answer 12:

(i) Required expression = (4 - 5x + 6x² - 8x³) - (6x³ - 7x² + 5x - 3)
                                      = 4 - 5x + 6x² - 8x³ - 6x³ + 7x² - 5x + 3
                                      = - 8x³- 6x³ + 7x² + 6x²- 5x - 5x + 3 + 4
                                      = - 14x³ + 13x² - 10x +7

(ii) Required expression  = (5x² - y + z + 7) - (- x² - 3z)
                                        = 5x² - y + z + 7 + x² + 3z
                                        = 5x²+ x² - y + z + 3z + 7
                                        = 6x² - y + 4z + 7

(iii) Required expression = (y³ - 3xy² - 4x²y) - (x³ + 2x²y + 6xy² - y³)
                                         = y³ - 3xy² - 4x²y - x³ - 2x²y - 6xy² + y³
                                         = y³ + y³ - 3xy²- 6xy² - 4x²y - 2x²y - x³
                                         = 2y³- 9xy² - 6x²y - x³

Question 13:

From
(i) p³ − 4 + 3p², take away 5p² − 3p³ + p − 6
(ii) 7 + x − x², take away 9 + x + 3x² + 7x³
(iii) 1 − 5y², take away y³ + 7y² + y + 1
(iv) x³ − 5x² + 3x + 1, take away 6x² − 4x³ + 5 + 3x

Answer 13:

 (i) Required expression = (p³ - 4 + 3p²) - (5p² - 3p³ + p - 6)
                                        = p³ - 4 + 3p² - 5p² + 3p³ - p + 6
                                        = p³ + 3p³ + 3p² - 5p²- p - 4+ 6
                                        = 4p³ - 2p² - p + 2

(ii) Required expression = (7 + x - x²) - (9 + x + 3x² + 7x³)
                                        = 7 + x - x² - 9 - x - 3x² - 7x³
                                        = - 7x³- x² - 3x² + 7 - 9
                                        = - 7x³ - 4x² - 2

(iii) Required expression = (1 - 5y²) - (y³+ 7y² + y + 1)
                                         = 1 - 5y² - y³ - 7y² - y - 1
                                         = - y³- 5y² - 7y² - y
                                         = - y³- 12y² - y

(iv) Required expression = (x³ - 5x² + 3x + 1) - (6x² - 4x³ + 5 +3x)
                                         = x³ - 5x² + 3x + 1 - 6x² + 4x³ - 5 - 3x
                                         = x³ + 4x³ - 5x² - 6x² + 1 - 5
                                         = 5x³ - 11x² - 4

Question 14:

From the sum of 3x² − 5x + 2 and −5x² − 8x + 9 subtract 4x² − 7x + 9.

Answer 14:

Required expression = {(3x² - 5x + 2) + (- 5x² - 8x + 9)} - (4x² - 7x + 9)
                                  = {3x² - 5x + 2 - 5x² - 8x + 9} -  (4x² - 7x + 9)
                                  = {3x² - 5x² - 5x - 8x + 2 + 9} -  (4x² - 7x + 9)
                                  = {- 2x² - 13x +11} - (4x² - 7x + 9)
                                  = - 2x² - 13x + 11 - 4x² + 7x - 9
                                  = - 2x² - 4x² - 13x + 7x + 11 - 9
                                  = - 6x² - 6x + 2

Question 15:

Subtract the sum of 13x − 4y + 7z and −6z + 6x + 3y from the sum of 6x − 4y − 4z and 2x + 4y − 7.

Answer 15:

Sum of (13x - 4y + 7z) and ( - 6z + 6x + 3y)
= {(13x - 4y + 7z) + (- 6z + 6x + 3y)
={13x - 4y + 7z - 6z + 6x + 3y}
= {13x + 6x - 4y + 3y + 7z - 6z}
= 19x - y + z

Sum of (6x − 4y − 4z) and (2x + 4y − 7)
= (6x − 4y − 4z) + (2x + 4y − 7)
= 6x − 4y − 4z + 2x + 4y − 7
= 8x - 4z - 7

Now, required expression = {(8x - 4z - 7) - (19x - y + z)}
                                  = 8x - 4z - 7 - 19x + y - z
                                  = 8x - 19x + y - 4z - z - 7
                                  = - 11x + y - 5z - 7

Question 16:

From the sum of x² + 3y² − 6xy, 2x² − y² + 8xy, y² + 8 and x² − 3xy subtract −3x² + 4y² − xy + x − y + 3.

Answer 16:

Sum of (x² + 3y² - 6xy), (2x² - y² + 8xy), (y² + 8) and (x² - 3xy)
={(x² + 3y² - 6xy) + (2x² - y² + 8xy) + ( y² + 8) + (x² - 3xy)}
={x² + 3y² - 6xy + 2x² - y² + 8xy + y² + 8 + x² - 3xy}
= {x²+ 2x²+ x² + 3y²- y² + y²- 6xy + 8xy - 3xy + 8}
= 4x² + 3y² - xy + 8

Now, required expression = (4x² + 3y² - xy + 8) - (- 3x² + 4y² - xy + x - y + 3)
                                  = 4x² + 3y² - xy + 8 + 3x² - 4y² + xy - x + y - 3
                                  = 4x² + 3x²+ 3y²- 4y²- x + y - 3 + 8
                                  = 7x² - y²- x + y + 5

Question 17:

What should be added to xy − 3yz + 4zx to get 4xy − 3zx + 4yz + 7?

Answer 17:

The required expression can be got by subtracting xy - 3yz + 4zx from 4xy - 3zx + 4yz + 7.
Therefore, required expression = (4xy - 3zx + 4yz + 7) - (xy - 3yz + 4zx)
                                  = 4xy - 3zx + 4yz + 7 - xy + 3yz - 4zx
                                  = 4xy - xy - 3zx - 4zx + 4yz + 3yz + 7
                                  = 3xy - 7zx + 7yz + 7

Question 18:

What should be subtracted from x² − xy + y² −x + y + 3 to obtain −x² + 3y² − 4xy + 1?

Answer 18:

Let 'M' be the required expression. Then, we have
x² - xy + y² - x + y + 3 - M = - x² + 3y² - 4xy + 1
Therefore,
M = (x² - xy + y² - x + y + 3) - (- x² + 3y² - 4xy + 1)
    = x² - xy + y² - x + y + 3 + x² - 3y² + 4xy - 1
Collecting positive and negative like terms together, we get
x² + x²- xy + 4xy + y²- 3y² - x + y + 3 - 1
= 2x² + 3xy- 2y²- x + y + 2

Question 19:

How much is x − 2y + 3z greater than 3x + 5y − 7?

Answer 19:

Required expression  = (x - 2y + 3z) - (3x + 5y - 7)
                                   =  x - 2y + 3z - 3x - 5y + 7
Collecting positive and negative like terms together, we get
x - 3x - 2y - 5y + 3z + 7
= - 2x - 7y + 3z + 7

Question 20:

How much is x² − 2xy + 3y² less than 2x² − 3y² + xy?

Answer 20:

Required expression = (2x² - 3y² + xy) - (x² - 2xy + 3y²)
                                  = 2x² - 3y² + xy - x² + 2xy - 3y²
Collecting positive and negative like terms together, we get
2x² - x² - 3y² - 3y² + xy + 2xy
 = x² - 6y² + 3xy

Question 21:

How much does a² − 3ab + 2b² exceed 2a² − 7ab + 9b²?

Answer 21:

Required expression = (a² - 3ab + 2b²) - (2a² - 7ab + 9b²)
                                  = a² - 3ab + 2b² - 2a² + 7ab - 9b²
Collecting positive and negative like terms together, we get
                                  =  a² - 2a²  - 3ab + 7ab  + 2b² -  9b²   
                                  = - a² + 4ab - 7b²

Question 22:

What must be added to 12x³ − 4x² + 3x − 7 to make the sum x³ + 2x² − 3x + 2?

Answer 22:

Let 'M' be the required expression. Thus, we have
12x³ - 4x² + 3x - 7 + M = x³ + 2x² - 3x + 2
Therefore,
M = (x³ + 2x² - 3x + 2) - (12x³ - 4x² + 3x - 7)
    =  x³ + 2x² - 3x + 2 - 12x³ + 4x² - 3x + 7
Collecting positive and negative like terms together, we get
x³- 12x³ + 2x² + 4x² - 3x - 3x + 2 + 7
= - 11x³ + 6x² - 6x + 9

Question 23:

If P = 7x² + 5xy − 9y², Q = 4y² − 3x² − 6xy and R = −4x² + xy + 5y², show that P + Q + R = 0.

Answer 23:

We have
P + Q + R = (7x² + 5xy - 9y²) + (4y² - 3x² - 6xy) + (- 4x² + xy + 5y²)
                 = 7x² + 5xy - 9y² + 4y² - 3x² - 6xy - 4x² + xy + 5y²
Collecting positive and negative like terms together, we get
7x²- 3x² - 4x² + 5xy - 6xy + xy - 9y² + 4y² + 5y²
= 7x²- 7x² + 6xy - 6xy  - 9y² + 9y²
= 0

Question 24:

If P = a² − b² + 2ab, Q = a² + 4b² − 6ab, R = b² + b, S = a² − 4ab and T = −2a² + b² − ab + a. Find P + Q + R + S − T.

Answer 24:

 We have
P + Q + R + S - T = {(a² - b² + 2ab) + (a² + 4b² - 6ab) + (b² + b) + (a² - 4ab)} - (-2a² + b² - ab + a)
                             = {a² - b² + 2ab + a² + 4b² - 6ab + b² + b + a² - 4ab}- (- 2a² + b² - ab + a)
                             = {3a² + 4b² - 8ab + b } - (-2a² + b² - ab + a)
                             = 3a²+ 4b² - 8ab + b + 2a² - b² + ab - a
Collecting positive and negative like terms together, we get
3a² + 2a² + 4b² - b² - 8ab + ab - a + b
= 5a² + 3b²- 7ab - a + b