8类RD Sharma解–第9章一个变量中的线性方程–练习9.2 |套装2

第九章一个变量中的线性方程–练习9.2 |套装1

问题14.(1-2x)/ 7 –(2-3x)/ 8 = 3/2 + x / 4

解决方案：

(1-2x)/7 – (2-3x)/8 = 3/2 + x/4

First rearrange the equation

(1-2x)/7 – (2-3x)/8 – x/4 = 3/2

By taking LCM for 7, 8 and 4 which is 56

((1-2x)8 – (2-3x)7 – 14x)/56 = 3/2

(8 – 16x – 14 + 21x – 14x)/56 = 3/2

(-9x – 6)/56 = 3/2

After cross-multiplying

2(-9x-6) = 3(56)

-18x – 12 = 168

-18x = 168+12

-18x = 180

x = 180/-18

x = -10

Now verify the equation by putting x = -10

(1-2x)/7 – (2-3x)/8 = 3/2 + x/4

x = -10

(1-2(-10))/7 – (2-3(-10))/8 = 3/2 + (-10)/4

(1+20)/7 – (2+30)/8 = 3/2 – 5/2

21/7 – 32/8 = 3/2 – 5/2

3 – 4 = -2/2

-1 = -1

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题15(9x + 7)/ 2 –(x –(x-2)/ 7)= 36

解决方案：

(9x+7)/2 – (x – (x-2)/7) = 36

First simplify the given equation

(9x+7)/2 – (7x-x+2)/7 = 36

(9x+7)/2 – (6x+2)/7 = 36

By taking LCM for 2 and 7 is 14

(7(9x+7) – 2(6x+2))/14 = 36

(63x+49 – 12x – 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 – (x – (x-2)/7) = 36

(9x+7)/2 – (6x+2)/7 = 36

x = 9

(9(9)+7)/2 – (6(9)+2)/7 = 36

(81+7)/2 – (54+2)/7 = 36

88/2 – 56/7 = 36

44 – 8 = 36

36 = 36

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题16.0.18(5x-4)= 0.5x + 0.8

解决方案：

0.18(5x – 4) = 0.5x + 0.8

First rearrange the given equation

0.18(5x – 4) – 0.5x = 0.8

0.90x – 0.72 – 0.5x = 0.8

0.90x – 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 – 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

问题17.2 / 3x – 3 / 2x = 1/12

解决方案：

2/3x – 3/2x = 1/12

By taking LCM for 3x and 2x which is 6x

((2×2) – (3×3))/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 – 3/2x = 1/12

x = -10

2/3(-10) – 3/2(-10) = 1/12

-2/30 + 3/20 = 1/12

((-2×2) + (3×3))/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.

问题18.4x / 9 + 1/3 + 13x / 108 =(8x + 19)/ 18

解决方案：

4x/9 + 1/3 + 13x/108 = (8x+19)/18

First rearrange the given equation

4x/9 + 13x/108 – (8x+19)/18 = -1/3

By taking LCM for 9, 108 and 18 which is 108

((4x×12) + 13x×1 – (8x+19)6)/108 = -1/3

(48x + 13x – 48x – 114)/108 = -1/3

(13x – 114)/108 = -1/3

After cross-multiplying

(13x – 114)3 = -108

39x – 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

((8×6) + (1×6) + (13×1))/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.

问题19(45-2x)/ 15 –(4x + 10)/ 5 =(15-14x)/ 9

解决方案：

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

First rearranging the given equation

(45-2x)/15 – (4x+10)/5 – (15-14x)/9 = 0

By taking LCM for 15, 5 and 9 which is 45

((45-2x)3 – (4x+10)9 – (15-14x)5)/45 = 0

(135 – 6x – 36x – 90 – 75 + 70x)/45 = 0

(28x – 30)/45 = 0

After cross-multiplying

28x – 30 = 0

28x = 30

x = 30/28

x = 15/14

Now verify the equation by putting x = 15/14

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

x = 15/14

(45-2(15/14))/15 – (4(15/14) + 10)/5 = (15 – 14(15/14))/9

(45- 15/7)/15 – (30/7 + 10)/5 = (15-15)/9

300/105 – 100/35 = 0

(300-300)/105 = 0

0 = 0

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题20. 5(7x + 5)/ 3 – 23/3 = 13 –(4x-2)/ 3

解决方案：

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

First rearrange the given equation

(35x + 25)/3 + (4x – 2)/3 = 13 + 23/3

(35x + 25 + 4x – 2)/3 = (39+23)/3

(39x + 23)/3 = 62/3

After cross-multiplying

(39x + 23)3 = 62(3)

39x + 23 = 62

39x = 62 – 23

39x = 39

x = 1

Now verify the equation by putting x = 1

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

x = 1

(35x + 25)/3 – 23/3 = 13 – (4x-2)/3

(35+25)/3 – 23/3 = 13 – (4-2)/3

60/3 – 23/3 = 13 – 2/3

(60-23)/3 = (39-2)/3

37/3 = 37/3

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题21.(7x-1)/ 4-1/3(2x-(1-x)/ 2)= 10/3

解决方案：

(7x-1)/4 – 1/3(2x – (1-x)/2) = 10/3

when we expand the given equation,

(7x-1)/4 – (4x-1+x)/6 = 10/3

(7x-1)/4 – (5x-1)/6 = 10/3

By taking LCM for 4 and 6 is 24

((7x-1)6 – (5x-1)4)/24 = 10/3

(42x – 6 – 20x + 4)/24 = 10/3

(22x – 2)/24 = 10/3

After cross-multiplying

22x – 2 = 10(8)

22x – 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 – 1/3(2x – (1-x)/2) = 10/3

x = 41/11

(7x-1)/4 – (5x-1)/6 = 10/3

(7(41/11)-1)/4 – (5(41/11)-1)/6 = 10/3

(287/11 – 1)/4 – (205/11 – 1)/6 = 10/3

(287-11)/44 – (205-11)/66 = 10/3

276/44 – 194/66 = 10/3

69/11 – 97/33 = 10/3

((69×3) – (97×1))/33 = 10/3

(207 – 97)/33 = 10/3

110/33 = 10/3

10/3 = 10/3

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题22.0.5(x-0.4)/0.35-0.6(x-2.71)/0.42 = x + 6.1

解决方案：

0.5(x-0.4)/0.35 – 0.6(x-2.71)/0.42 = x + 6.1

First simplify the given equation

(0.5/0.35)(x – 0.4) – (0.6/0.42)(x – 2.71) = x + 6.1

(x – 0.4)/0.7 – (x – 2.71)/0.7 = x + 6.1

(x – 0.4 – x + 2.71)/0.7 = x + 6.1

-0.4 + 2.71 = 0.7(x + 6.1)

0.7x = 2.71 – 0.4 – 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 – 0.6(x-2.71)/0.42 = x + 6.1

x = -2.8

0.5(-2.8 – 0.4)/0.35 – 0.6(-2.8 – 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.

问题23：6.5x +(19.5x – 32.5)/ 2 = 6.5x + 13 +(13x – 26)/ 2

解决方案：

6.5x + (19.5x – 32.5)/2 = 6.5x + 13 + (13x – 26)/2

First rearrange the equation

6.5x + (19.5x – 32.5)/2 – 6.5x – (13x – 26)/2 = 13

(19.5x – 32.5)/2 – (13x – 26)/2 = 13

(19.5x – 32.5 – 13x + 26)/2 = 13

(6.5x – 6.5)/2 = 13

6.5x – 6.5 = 13×2

6.5x – 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 – 32.5)/2 = 6.5x + 13 + (13x – 26)/2

x= 5

6.5(5) + (19.5(5) – 32.5)/2 = 6.5(5) + 13 + (13(5) – 26)/2

32.5 + (97.5 – 32.5)/2 = 32.5 + 13 + (65 – 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.

问题24.(3x – 8)(3x + 2)–(4x – 11)(2x +1)=(x – 3)(x + 7)

解决方案：

(3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

First simplify the given equation

9x² + 6x – 24x – 16 – 8x² – 4x + 22x + 11 = x² + 7x – 3x – 21

9x² + 6x – 24x – 16 – 8x² – 4x + 22x + 11 – x² – 7x + 3x + 21 = 0

9x² – 8x² – x² + 6x – 24x – 4x + 22x – 7x + 3x – 16 + 21 + 11 = 0

-4x + 16 = 0

-4x = -16

x = 4

Now verify the equation by putting x = 4

(3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

x = 4

(3(4) – 8) (3(4) + 2) – (4(4) – 11) (2(4) + 1) = (4 – 3) (4 + 7)

(12-8) (12+2) – (16-11) (8+1) = 1(11)

4 (14) – 5(9) = 11

56 – 45 = 11

11 = 11

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

问题25. [(2x + 3)+(x + 5)] ² + [(2x + 3)–(x + 5)] ² = 10x ² + 92

解决方案：

[(2x+3) + (x+5)]- + [(2x+3) – (x+5)]² = 10x² + 92

First simplify the given equation 

[3x + 8]² + [x – 2]² = 10x² + 92

Now apply the formula (a+b)2

9x² + 48x + 64 + x² – 4x + 4 = 10x² + 92

After rearranging the equation

9x² – 10x² + x² + 48x – 4x = 92 – 64 – 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) – (x+5)]² = 10x² + 92

x = 6/11

[2(6/11) + 3 + (6/11) + 5]² + [2(6/11) + 3 – (6/11) – 5]² = 10(6/11)² + 92

[(12/11 + 3) + (6/11 + 5)]² + [(12/11 + 3) – (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) – (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.