Class 8 RD Sharma Solutions â Chapter 9 Linear Equation In One Variable â Exercise 9.2 | Set 2
Question 14. (1-2x)/7 â (2-3x)/8 = 3/2 + x/4
Solution:
(1-2x)/7 â (2-3x)/8 = 3/2 + x/4
First rearrange the equation
(1-2x)/7 â (2-3x)/8 â x/4 = 3/2
By taking LCM for 7, 8 and 4 which is 56
((1-2x)8 â (2-3x)7 â 14x)/56 = 3/2
(8 â 16x â 14 + 21x â 14x)/56 = 3/2
(-9x â 6)/56 = 3/2
After cross-multiplying
2(-9x-6) = 3(56)
-18x â 12 = 168
-18x = 168+12
-18x = 180
x = 180/-18
x = -10
Now verify the equation by putting x = -10
(1-2x)/7 â (2-3x)/8 = 3/2 + x/4
x = -10
(1-2(-10))/7 â (2-3(-10))/8 = 3/2 + (-10)/4
(1+20)/7 â (2+30)/8 = 3/2 â 5/2
21/7 â 32/8 = 3/2 â 5/2
3 â 4 = -2/2
-1 = -1
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 15. (9x+7)/2 â (x â (x-2)/7) = 36
Solution:
(9x+7)/2 â (x â (x-2)/7) = 36
First simplify the given equation
(9x+7)/2 â (7x-x+2)/7 = 36
(9x+7)/2 â (6x+2)/7 = 36
By taking LCM for 2 and 7 is 14
(7(9x+7) â 2(6x+2))/14 = 36
(63x+49 â 12x â 4)/14 = 36
(51x + 45)/14 = 36
After cross-multiplying
51x + 45 = 36(14)
51x + 45 = 504
51x = 504-45
51x = 459
x = 459/51
x = 9
Now verify the equation by putting x = 9
(9x+7)/2 â (x â (x-2)/7) = 36
(9x+7)/2 â (6x+2)/7 = 36
x = 9
(9(9)+7)/2 â (6(9)+2)/7 = 36
(81+7)/2 â (54+2)/7 = 36
88/2 â 56/7 = 36
44 â 8 = 36
36 = 36
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 16. 0.18(5x â 4) = 0.5x + 0.8
Solution:
0.18(5x â 4) = 0.5x + 0.8
First rearrange the given equation
0.18(5x â 4) â 0.5x = 0.8
0.90x â 0.72 â 0.5x = 0.8
0.90x â 0.5x = 0.8 + 0.72
0.40x = 1.52
x = 1.52/0.40
x = 3.8
Now verify the equation by putting x = 3.8
0.18(5x â 4) = 0.5x + 0.8
x = 3.8
0.18(5(3.8)-4) = 0.5(3.8) + 0.8
0.18(19-4) = 1.9 + 0.8
2.7 = 2.7
Thus L.H.S. = R.H.S.,
Hence, the equation is verified
Question 17. 2/3x â 3/2x = 1/12
Solution:
2/3x â 3/2x = 1/12
By taking LCM for 3x and 2x which is 6x
((2Ă2) â (3Ă3))/6x = 1/12
(4-9)/6x = 1/12
-5/6x = 1/12
After cross-multiplying
6x = -60
x = -60/6
x = -10
Now verify the equation by putting x = -10
2/3x â 3/2x = 1/12
x = -10
2/3(-10) â 3/2(-10) = 1/12
-2/30 + 3/20 = 1/12
((-2Ă2) + (3Ă3))/60 = 1/12
(-4+9)/60 = 1/12
5/60 = 1/12
1/12 = 1/12
Thus L.H.S. = R.H.S.,
Hence the equation is verified.
Question 18. 4x/9 + 1/3 + 13x/108 = (8x+19)/18
Solution:
4x/9 + 1/3 + 13x/108 = (8x+19)/18
First rearrange the given equation
4x/9 + 13x/108 â (8x+19)/18 = -1/3
By taking LCM for 9, 108 and 18 which is 108
((4xĂ12) + 13xĂ1 â (8x+19)6)/108 = -1/3
(48x + 13x â 48x â 114)/108 = -1/3
(13x â 114)/108 = -1/3
After cross-multiplying
(13x â 114)3 = -108
39x â 342 = -108
39x = -108 + 342
39x = 234
x = 234/39
x = 6
Now verify the equation by putting x = 6
4x/9 + 1/3 + 13x/108 = (8x+19)/18
x = 6
4(6)/9 + 1/3 + 13(6)/108 = (8(6)+19)/18
24/9 + 1/3 + 78/108 = 67/18
8/3 + 1/3 + 13/18 = 67/18
((8Ă6) + (1Ă6) + (13Ă1))/18 = 67/18
(48 + 6 + 13)/18 = 67/18
67/18 = 67/18
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 19. (45-2x)/15 â (4x+10)/5 = (15-14x)/9
Solution:
(45-2x)/15 â (4x+10)/5 = (15-14x)/9
First rearranging the given equation
(45-2x)/15 â (4x+10)/5 â (15-14x)/9 = 0
By taking LCM for 15, 5 and 9 which is 45
((45-2x)3 â (4x+10)9 â (15-14x)5)/45 = 0
(135 â 6x â 36x â 90 â 75 + 70x)/45 = 0
(28x â 30)/45 = 0
After cross-multiplying
28x â 30 = 0
28x = 30
x = 30/28
x = 15/14
Now verify the equation by putting x = 15/14
(45-2x)/15 â (4x+10)/5 = (15-14x)/9
x = 15/14
(45-2(15/14))/15 â (4(15/14) + 10)/5 = (15 â 14(15/14))/9
(45- 15/7)/15 â (30/7 + 10)/5 = (15-15)/9
300/105 â 100/35 = 0
(300-300)/105 = 0
0 = 0
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 20. 5(7x + 5)/3 â 23/3 = 13 â (4x-2)/3
Solution:
5(7x+5)/3 â 23/3 = 13 â (4x-2)/3
First rearrange the given equation
(35x + 25)/3 + (4x â 2)/3 = 13 + 23/3
(35x + 25 + 4x â 2)/3 = (39+23)/3
(39x + 23)/3 = 62/3
After cross-multiplying
(39x + 23)3 = 62(3)
39x + 23 = 62
39x = 62 â 23
39x = 39
x = 1
Now verify the equation by putting x = 1
5(7x+5)/3 â 23/3 = 13 â (4x-2)/3
x = 1
(35x + 25)/3 â 23/3 = 13 â (4x-2)/3
(35+25)/3 â 23/3 = 13 â (4-2)/3
60/3 â 23/3 = 13 â 2/3
(60-23)/3 = (39-2)/3
37/3 = 37/3
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 21. (7x-1)/4 â 1/3(2x â (1-x)/2) = 10/3
Solution:
(7x-1)/4 â 1/3(2x â (1-x)/2) = 10/3
when we expand the given equation,
(7x-1)/4 â (4x-1+x)/6 = 10/3
(7x-1)/4 â (5x-1)/6 = 10/3
By taking LCM for 4 and 6 is 24
((7x-1)6 â (5x-1)4)/24 = 10/3
(42x â 6 â 20x + 4)/24 = 10/3
(22x â 2)/24 = 10/3
After cross-multiplying
22x â 2 = 10(8)
22x â 2 = 80
22x = 80+2
22x = 82
x = 82/22
x = 41/11
Now verify the equation by putting x = 41/11
(7x-1)/4 â 1/3(2x â (1-x)/2) = 10/3
x = 41/11
(7x-1)/4 â (5x-1)/6 = 10/3
(7(41/11)-1)/4 â (5(41/11)-1)/6 = 10/3
(287/11 â 1)/4 â (205/11 â 1)/6 = 10/3
(287-11)/44 â (205-11)/66 = 10/3
276/44 â 194/66 = 10/3
69/11 â 97/33 = 10/3
((69Ă3) â (97Ă1))/33 = 10/3
(207 â 97)/33 = 10/3
110/33 = 10/3
10/3 = 10/3
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 22. 0.5(x-0.4)/0.35 â 0.6(x-2.71)/0.42 = x + 6.1
Solution:
0.5(x-0.4)/0.35 â 0.6(x-2.71)/0.42 = x + 6.1
First simplify the given equation
(0.5/0.35)(x â 0.4) â (0.6/0.42)(x â 2.71) = x + 6.1
(x â 0.4)/0.7 â (x â 2.71)/0.7 = x + 6.1
(x â 0.4 â x + 2.71)/0.7 = x + 6.1
-0.4 + 2.71 = 0.7(x + 6.1)
0.7x = 2.71 â 0.4 â 4.27
= -1.96
x = -1.96/0.7
x = -2.8
Now verify the equation by putting x = 5
0.5(x-0.4)/0.35 â 0.6(x-2.71)/0.42 = x + 6.1
x = -2.8
0.5(-2.8 â 0.4)/0.35 â 0.6(-2.8 â 2.71)/0.42 = -2.8 + 6.1
-1.6/0.35 + 3.306/0.42 = 3.3
-4.571 + 7.871 = 3.3
3.3 = 3.3
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 23. 6.5x + (19.5x â 32.5)/2 = 6.5x + 13 + (13x â 26)/2
Solution:
6.5x + (19.5x â 32.5)/2 = 6.5x + 13 + (13x â 26)/2
First rearrange the equation
6.5x + (19.5x â 32.5)/2 â 6.5x â (13x â 26)/2 = 13
(19.5x â 32.5)/2 â (13x â 26)/2 = 13
(19.5x â 32.5 â 13x + 26)/2 = 13
(6.5x â 6.5)/2 = 13
6.5x â 6.5 = 13Ă2
6.5x â 6.5 = 26
6.5x = 26+6.5
6.5x = 32.5
x = 32.5/6.5
x = 5
Now verify the equation by putting x = 5
6.5x + (19.5x â 32.5)/2 = 6.5x + 13 + (13x â 26)/2
x= 5
6.5(5) + (19.5(5) â 32.5)/2 = 6.5(5) + 13 + (13(5) â 26)/2
32.5 + (97.5 â 32.5)/2 = 32.5 + 13 + (65 â 26)/2
32.5 + 65/2 = 45.5 + 39/2
(65 + 65)/2 = (91+39)/2
130/2 = 130/2
65 = 65
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 24. (3x â 8) (3x + 2) â (4x â 11) (2x + 1) = (x â 3) (x + 7)
Solution:
(3x â 8) (3x + 2) â (4x â 11) (2x + 1) = (x â 3) (x + 7)
First simplify the given equation
9x2 + 6x â 24x â 16 â 8x2 â 4x + 22x + 11 = x2 + 7x â 3x â 21
9x2 + 6x â 24x â 16 â 8x2 â 4x + 22x + 11 â x2 â 7x + 3x + 21 = 0
9x2 â 8x2 â x2 + 6x â 24x â 4x + 22x â 7x + 3x â 16 + 21 + 11 = 0
-4x + 16 = 0
-4x = -16
x = 4
Now verify the equation by putting x = 4
(3x â 8) (3x + 2) â (4x â 11) (2x + 1) = (x â 3) (x + 7)
x = 4
(3(4) â 8) (3(4) + 2) â (4(4) â 11) (2(4) + 1) = (4 â 3) (4 + 7)
(12-8) (12+2) â (16-11) (8+1) = 1(11)
4 (14) â 5(9) = 11
56 â 45 = 11
11 = 11
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
Question 25. [(2x+3) + (x+5)]2 + [(2x+3) â (x+5)]2 = 10x2 + 92
Solution:
[(2x+3) + (x+5)]- + [(2x+3) â (x+5)]2 = 10x2 + 92
First simplify the given equation
[3x + 8]2 + [x â 2]2 = 10x2 + 92
Now apply the formula (a+b)2
9x2 + 48x + 64 + x2 â 4x + 4 = 10x2 + 92
After rearranging the equation
9x2 â 10x2 + x2 + 48x â 4x = 92 â 64 â 4
44x = 24
x = 24/44
x = 6/11
Now verify the equation by putting x = 6/11
[(2x+3) + (x+5)]2 + [(2x+3) â (x+5)]2 = 10x2 + 92
x = 6/11
[2(6/11) + 3 + (6/11) + 5]2 + [2(6/11) + 3 â (6/11) â 5]2 = 10(6/11)2 + 92
[(12/11 + 3) + (6/11 + 5)]2 + [(12/11 + 3) â (6/11 + 5)]2 = 10(6/11)2 + 92
[(12+33)/11 + (6+55)/11]2 + [(12+33)/11- (6+55)/11]2 = 10(6/11)2 + 92
[(45/11)+ (61/11)]2 + [(45/11) â (61/11)]2 = 360/121 + 92
(106/11)2 + (-16/11)2 = (360 + 11132)/121
11236/121 + 256/121 = 11492/121
11492/121 = 11492/121
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.