求解以下每个方程式，并检查每种情况下的结果：
问题1.(2x + 5)/ 3 = 3x – 10
解决方案：
First simplify the equation,
(2x + 5)/3 – 3x = – 10
By taking LCM
(2x + 5 – 9x)/3 = -10
(-7x + 5)/3 = -10
After cross-multiplication we get,
-7x + 5 = -30
-7x = -30 – 5
-7x = -35
x = -35/-7
= 5
Now verify the equation by putting x = 5.
(2x + 5)/3 = 3x – 10
x = 5
(2×5 + 5)/3 = 3(5) – 10
(10+5)/3 = 15-10
15/3 = 5
5 = 5
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题2.(a – 8)/ 3 =(a – 3)/ 2
解决方案：
After cross-multiplication we will get,
(a – 8)2 = (a-3)3
2a – 16 = 3a – 9
2a – 3a = -9 + 16
-a = 7
a = -7
Now verify the equation by putting a = -7.
(a – 8)/3 = (a – 3)/2
a = -7
(-7 – 8)/3 = (-7 – 3)/2
-15/3 = -10/2
-5 = -5
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题3.(7y + 2)/ 5 =(6y – 5)/ 11
解决方案：
After cross-multiplication we will get,
(7y + 2)11 = (6y – 5)5
77y + 22 = 30y – 25
77y – 30y = -25 – 22
47y = -47
y = -47/47
y = -1
Now verify the equation by putting y = -1.
(7y + 2)/5 = (6y – 5)/11
y =-1
(7(-1) + 2)/5 = (6(-1) – 5)/11
(-7 + 2)/5 = (-6 – 5)/11
-5/5 = -11/11
-1 = -1
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题4. x – 2x + 2 – 16 / 3x + 5 = 3 – 7 / 2x
解决方案：
x – 2x + 2 – 16/3x + 5 = 3 – 7/2x
First rearrange the equation
x – 2x – 16x/3 + 7x/2 = 3 – 2 – 5
By taking LCM for 2 and 3 which is 6
(6x – 12x – 32x + 21x)/6 = -4
-17x/6 = -4
After cross-multiplying we will get,
-17x = -4×6
-17x = -24
x = -24/-17
x = 24/17
Now verify the equation by putting x = 24/17.
x – 2x + 2 – 16/3x + 5 = 3 – 7/2x
x = 24/17
24/17 – 2(24/17) + 2 – (16/3)(24/17) + 5 = 3 – (7/2)(24/17)
24/17 – 48/17 + 2 – 384/51 + 5 = 3 – 168/34
By taking 51 and 17 as the LCM we get,
(72 – 144 + 102 – 384 + 255)/51 = (102 – 168)/34
-99/51 = -66/34
-33/17 = -33/17
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题5. 1 / 2x + 7x – 6 = 7x + 1/4
解决方案：
1/2x + 7x – 6 = 7x + 1/4
First rearrange the equation
1/2x + 7x – 7x = 1/4 + 6 (by taking LCM)
1/2x = (1+ 24)/4
1/2x = 25/4
After cross-multiplying
4x = 25 × 2
4x = 50
x = 50/4
x = 25/2
Now verify the equation by putting x = 25/2.
1/2x + 7x – 6 = 7x + 1/4
x = 25/2
(1/2) (25/2) + 7(25/2) – 6 = 7(25/2) + 1/4
25/4 + 175/2 – 6 = 175/2 + 1/4
By taking LCM for 4 and 2 is 4
(25 + 350 – 24)/4 = (350+1)/4
351/4 = 351/4
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题6. 3 / 4x + 4x = 7/8 + 6x – 6
解决方案：
3/4x + 4x = 7/8 + 6x – 6
First rearrange the equation
3/4x + 4x – 6x = 7/8 – 6
By taking 4 and 8 as LCM
(3x + 16x – 24x)/4 = (7 – 48)/8
-5x/4 = -41/8
After cross-multiplying
-5x(8) = -41(4)
-40x = -164
x = -164/-40
= 82/20
= 41/10
Now verify the equation by putting x = 41/10.
3/4x + 4x = 7/8 + 6x – 6
x = 41/10
(3/4)(41/10) + 4(41/10) = 7/8 + 6(41/10) – 6
123/40 + 164/10 = 7/8 + 246/10 – 6
(123 + 656)/40 = (70 + 1968 – 480)/80
779/40 = 1558/80
779/40 = 779/40
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题7. 7x / 2 – 5x / 2 = 20x / 3 + 10
解决方案：
7x/2 – 5x/2 = 20x/3 + 10
First rearrange the equation
7x/2 – 5x/2 – 20x/3 = 10
By taking LCM for 2 and 3 is 6
(21x – 15x – 40x)/6 = 10
-34x/6 = 10
After cross-multiplying
-34x = 60
x = 60/-34
= -30/17
Now verify the equation by putting x = -30/7.
7x/2 – 5x/2 = 20x/3 + 10
x = -30/7
(7-/2)(-30/17) – (5/2)(-30/17) = (20/3)(-30/17) + 10
-210/34 +150/34 = -600/51 + 10
-30/17 = (-600+510)/51
= -90/51
-30/17 = -30/17
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题8.(6x +1)/ 2 +1 =(7x – 3)/ 3
解决方案：
(6x+1)/2 + 1 = (7x-3)/3
(6x + 1 + 2)/2 = (7x – 3)/3
After cross-multiplying
(6x + 3)3 = (7x – 3)2
18x + 9 = 14x – 6
18x – 14x = -6 – 9
4x = -15
x = -15/4
Now verify the equation by putting x = -15/4.
(6x+1)/2 + 1 = (7x-3)/3
x = -15/4
(6(-15/4) + 1)/2 + 1 = (7(-15/4) – 3)/3
(3(-15/2) + 1)/2 + 1 = (-105/4 -3)/3
(-45/2 + 1)/2 + 1 = (-117/4)/3
(-43/4) + 1 = -117/12
(-43+4)/4 = -39/4
-39/4 = -39/4
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题9.(3a-2)/ 3 +(2a + 3)/ 2 = a + 7/6
解决方案：
(3a-2)/3 + (2a+3)/2 = a + 7/6
First rearrange the equation
(3a-2)/3 + (2a+3)/2 – a = 7/6
By taking LCM for 2 and 3 which is 6
((3a-2)2 + (2a+3)3 – 6a)/6 = 7/6
(6a – 4 + 6a + 9 – 6a)/6 = 7/6
(6a + 5)/6 = 7/6
6a + 5 = 7
6a = 7-5
6a = 2
a = 2/6
a = 1/3
Now verify the equation by putting a = 1/3.
(3a-2)/3 + (2a+3)/2 = a + 7/6
a = 1/3
(3(1/3)-2)/3 + (2(1/3) + 3)/2 = 1/3 + 7/6
(1-2)/3 + (2/3 + 3)/2 = (2+7)/6
-1/3 + (11/3)/2 = 9/6
-1/3 + 11/6 = 3/2
(-2+11)/6 = 3/2
9/6 = 3/2
3/2 = 3/2
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题10. x –(x – 1)/ 2 = 1 –(x – 2)/ 3
解决方案：
x – (x-1)/2 = 1 – (x-2)/3
First rearrange the equation
x – (x-1)/2 + (x-2)/3 = 1
By taking LCM for 2 and 3 which is 6
(6x – (x-1)3 + (x-2)2)/6 = 1
(6x – 3x + 3 + 2x – 4)/6 = 1
(5x – 1)/6 = 1
After cross-multiplying
5x – 1 = 6
5x = 6 + 1
x = 7/5
Now verify the equation by putting x = 7/5.
x – (x-1)/2 = 1 – (x-2)/3
x = 7/5
7/5 – (7/5 – 1)/2 = 1 – (7/5 – 2)/3
7/5 – (2/5)/2 = 1 – (-3/5)/3
7/5 – 2/10 = 1 + 3/15
(14 – 2)/10 = (15+3)/15
12/10 = 18/15
6/5 = 6/5
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题11。3x / 4 –(x-1)/ 2 =(x-2)/ 3
解决方案：
3x/4 – (x-1)/2 = (x-2)/3
First rearrange the equation
3x/4 – (x-1)/2 – (x-2)/3 = 0
By taking LCM for 4, 2 and 3 which is 12
(9x – (x-1)6 – (x-2)4)/12 = 0
(9x – 6x + 6 – 4x + 8)/12 = 0
(-x + 14)/12 = 0
After cross-multiplying
-x + 14 = 0
x = 14
Now verify the equation by putting x = 14
3x/4 – (x-1)/2 = (x-2)/3
x = 14
3(14)/4 – (14-1)/2 = (14-2)/3
42/4 – 13/2 = 12/3
(42 – 26)/4 = 4
16/4 = 4
4 = 4
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题12：5x / 3 –(x-1)/ 4 =(x-3)/ 5
解决方案：
5x/3 – (x-1)/4 = (x-3)/5
First rearrange the equation
5x/3 – (x-1)/4 – (x-3)/5 = 0
By taking LCM for 3, 4 and 5 which is 60
((5x × 20) – (x-1)15 – (x-3)12)/60 = 0
(100x – 15x + 15 -12x + 36)/60 = 0
(73x + 51)/60 = 0
After cross-multiplying
73x + 51 = 0
x = -51/73
Now verify the equation by putting x = -51/73
5x/3 – (x-1)/4 = (x-3)/5
x = -51/73
(20x – (x-1)3)/12 = (-51/73 – 3)/5
(20x – 3x + 3)/12 = (-270/73)/5
(17x + 3)/12 = -270/365
(17(-51/73) + 3)/12 = -54/73
(-867/73 + 3)/12 = -54/73
((-867 + 219)/73)/12 = -54/73
(-648)/876 = -54/73
-54/73 = -54/73
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.
问题13(3x + 1)/ 16 +(2x-3)/ 7 =(x + 3)/ 8 +(3x-1)/ 14
解决方案：
(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14
First rearrange the equation
(3x+1)/16 + (2x-3)/7 – (x+3)/8 – (3x-1)/14 = 0
By taking LCM for 16, 7, 8 and 14 which is 112
((3x+1)7 + (2x-3)16 – (x+3)14 – (3x-1)8)/112 = 0
(21x + 7 + 32x – 48 – 14x – 42 – 24x + 8)/112 = 0
(21x + 32x – 14x – 24x + 7 – 48 – 42 + 8)/112 = 0
(15x – 75)/112 = 0
After cross-multiplying
15x – 75 = 0
15x = 75
x = 75/15
x = 5
Now verify the equation by putting x = 5
(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14
x = 5
(3(5)+1)/16 + (2(5)-3)/7 = (5+3)/8 + (3(5)-1)/14
(15+1)/16 + (10-3)/7 = 8/8 + (15-1)/14
16/16 + 7/7 = 8/8 + 14/14
1 + 1 = 1 + 1
2 = 2
Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.