Exercise 7.2
1. Add the following:
(i) 3x and 7x
Solution:
We have
3x + 7x = (3 + 7) x = 10x
(ii) -5xy and 9xy
Solution:
We have
-5xy + 9xy = (-5 + 9)xy = 4xy
2. Simplify each of the following:
(i) 7x3y +9yx3
Solution:
Given
7x3y+9yx3
7x3y+9yx3 = (7+9) x3y
=> 16x3y
(ii) 12a2b + 3ba2
Solution:
Given
12a2+3ba2= (12+3) a2b
=> 15a2b
3. Add the following:
Adding the given terms, we have
(i) 7abc, -5abc, 9abc, -8abc
Solution:
7abc + (-5abc) + (9abc) + (-8abc)
= 7abc – 5abc + 9abc – 8abc
= (7 – 5 + 9 – 8)abc
= (16 – 13)abc
= 3abc
(ii) 2x2y, – 4x2y, 6x2y, -5x2y
Solution:
2x2y, – 4x2y, 6x2y, -5x2y
= 2x2y – 4x2y + 6x2y – 5x2y
= (2- 4 + 6 – 5) x 2y
= (8 – 9) x 2y
= -x2y
4. Add the following expressions:
Solution:
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
5. Add the following expressions:
(i) 8a – 6ab + 5b, –6a – ab – 8b and –4a + 2ab + 3b
Solution:
Given 8a – 6ab +5b, – 6a-8b and -4a+2ab+3b
= (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) 5x3 + 7 + 6x – 5x2, 2x2 – 8 – 9x, 4x – 2x2 + 3 x 3, 3 x 3 – 9x – x2 and x – x2 – x3 – 4
Solution:
Given 5x3 + 7 + 6x – 5x2, 2x2 – 8 – 9x, 4x – 2x2 + 3 x 3, 3 x 3 – 9x – x2 and x – x2 – x3 – 4
= (53+7+6x-5x2)+(22-8-9x) + (4x-2x2+33) + (33-9x-x2) + (x-x2-x3-4)
= Collecting positive and negative like terms together, we get
6. Add the following:
(i) x – 3y – 2z
5x + 7y – 8z
3x – 2y + 5z
(ii) 4ab – 5bc + 7ca
–3ab + 2bc – 3ca
5ab – 3bc + 4ca
Solution:
(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 + 11ca – 3ca
= 6ab – 6bc + 8ca
7. Add 2×2 – 3x + 1 to the sum of 3×2 – 2x and 3x + 7.
Solution:
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
8. Add x2 + 2xy + y2 to the sum of x2 – 3y2and 2x2 – y2 + 9.
Solution:
9. Add a3+ b3 – 3 to the sum of 2a3 – 3b3 – 3ab + 7 and -a3 + b3 + 3ab – 9.
Solution:
10. Subtract:
(i) 7a2b from 3a2b
Solution:
Given 7a2b from 3a2b
= 3a2b-7a2b
= (3-7) a2b
= -4a2b
(ii) 4xy from -3xy
Solution:
Given 4xy from -3xy
= -3xy-4xy
= -7xy
11. Subtract:
(i) – 4x from 3y
Solution:
Required expression = (3y) – (–4x)
= 3y + 4x
(ii) – 2x from – 5y
Solution:
Required expression = (-5y) – (–2x)
12. Subtract:
(i) 6x3 −7x2 + 5x − 3 from 4 − 5x + 6x2 − 8x3
Solution:
(ii) − x2 −3z from 5x2 – y + z + 7
Solution:
(iii) x3 + 2x2y + 6xy2 − y3 from y3−3xy2−4x2y = –5y + 2x
Solution:
13. From:
(i) p3 – 4 + 3p2, take away 5p2 − 3p3 + p − 6
Solution:
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) 7 + x − x2, take away 9 + x + 3x2 + 7x3
Solution:
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) 1− 5y2, take away y3 + 7y2 + y + 1
Solution:
Required expression = (1− 5y2) – (y3 + 7y2 + y + 1)
= 1 – 5y2 – y3 – 7y2 – y – 1
= – y3 – 5y2 – 7y2 – y
= -= y3 – 12y2 – y
(iv) x3 − 5x2 + 3x + 1, take away 6x2 − 4x3 + 5 + 3x
Solution:
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
14. From the sum of 3x2 − 5x + 2 and − 5x2 − 8x + 9 subtract 4x2 − 7x + 9.
Solution:
Firstly, we have to add 3x2 − 5x + 2 and − 5x2 − 8x + 9 then from the result we have to subtract 4x2 − 7x + 9
= {(3x2 – 5x + 2) + (-5x2 – 8x + 9 )} – (4x2 – 7x+9)
= { 3x2 – 5x +2 -5x2 – 8x + 9 } – (4x2 – 7x +9 )
= { 3x2– 5x + 2 – 5x2– 8x + 9 } – (4x2-7x+9)
= {-2x2 – 13x + 11} – (4x2-7x +9)
= -2x2 – 13x+ 11-4x2+7x-9
= – 2x2 – 4x2 – 13x+7x+11- 9
= – 6x21- 6x+2
15. Subtract the sum of 13x – 4y + 7z and – 6z + 6x + 3y from the sum of 6x – 4y – 4z and 2x + 4y – 7.
Solution:
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)
= (6x + 2x – 4z – 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
16. From the sum of x2 + 3y2 − 6xy, 2x2 − y2 + 8xy, y2 + 8 and x2 − 3xy subtract −3x2 + 4y2 – xy + x – y + 3.
Solution:
First we have to find the 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+y2-6xy + 8xy – 3xy + 8
= 4x2 + 3y2 – xy + 8
Now, from the result subtract the -3×2+4y2-xy + x- y +3.
Therefore, required expression = (4x2+3y2-xy+8) – (-3x2
+ 4y2 – xy + x – y + 3)
= 4x2 + 3x2 + 3y2 – 4y2 – x + y3+ 8
= 7x2 – y2 x + y + 5
17. What should be added to xy – 3yz + 4zx to get 4xy – 3zx + 4yz + 7?
Solution:
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
18. What should be subtracted from x2 – xy + y2 – x + y + 3 to obtain −x2 + 3y2 − 4xy + 1?
Solution:
=> x2 – xy + y2 – x + y + 3 + x2 – 3y2 + 4xy -1
=> (1+1) x2 + (4-1) xy + (1-3) y2– – x + y +2
=> 2x2 + 3xy – 2y2 – x + y + 2
19. How much is x – 2y + 3z greater than 3x + 5y – 7?
Solution:
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
20. How much is x2 − 2xy + 3y2 less than 2x2 − 3y2 + xy?
Solution:
By subtracting the x2 – 2xy + 3y2 from 2×2 – 3y2 – xy we can get the required expression,
Required expression = (2×2- 3y2 + xy) – (x2 – 2xy + 3y2)
= 2x2 – 3y2 + xy = x2 + 2xy – 3y2
Collecting positive and negative like terms together, we get
= 2x2 – x2 – 3y2 + xy + 2xy
= x2 – 6y2 + 3xy
21. How much does a2 − 3ab + 2b2 exceed 2a2 − 7ab + 9b2?
Solution:
Required expression = a2 – 3ab + 2b2 – 2a2 – 7ab – 9b2
= (1 – 2) a2 + (- 3+7) ab + (2 -9) b2
= – a2 + 4ab – 7b2
Requred expression is – a2+ 4ab – 7b2
22. What must be added to 12x3 − 4x2 + 3x − 7 to make the sum x3 + 2x2 − 3x + 2?
Solution:
Let ‘E’ be the required expression. Thus, we have
12x3 – 4x2 + 3x – 7 + E = x3 + 2x2 – 3x+2
Therefore, E = (x3+ 12x2—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 – 3x + 2 + 7
= – 11x3 + 6x2 – 6x + 9
23. If P = 7x2 + 5xy − 9y2, Q = 4y2 − 3x2 − 6xy and R = −4x2 + xy + 5y2, show that P + Q + R = 0.
Solution:
Required expression = P + Q + R
= P + Q + R = 7×2 + 5xy – 9y2 + 4y2 – 3×2 – 6xy – 4×2 + xy
= (7 – 3 – 4) x2 + (5 – 6 + 1 ) xy + (5 + 4- 9) y2
= 0 + 0 + 0
= 0
∴ Hence proved
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.
Solution:
P + Q + R + S – T = a2-b2+a2+4b2 – 4b2 – 4ab + b2 + b + a2 – 4b + 2a2– b2 + ab – a
= (1+1+1+2) a2 + (-1+ 4+ 1 – 1) b2
+ (- 6 – 4 + 1) ab – a + b
= 5a2 + 3b2 – 7ab – a + b.