解决以下方程组：
问题14.0.5x + 0.7y = 0.74和0.3x + 0.5y = 0.5
解决方案：
0.5x + 0.7y = 0.74……………………… (i)
0.3x – 0.5y = 0.5 ………………………….. (ii)
Multiply LHS and RHS by 100 in (i) and (ii)
50x +70y = 74 ……………………….. (iii)
30x + 50y = 50 …………………………… (iv)
From (iii)
50x = 74 – 70y
x = (74−70y) / 50 ……………………………… (v)
Substituting x in equation (iv)
30[(74−70y)/ 50] + 50y = 50
Taking 50 as LCM
⇒ 2220 – 2100y + 2500y = 2500
Dividing by 10
⇒ 222 – 210y + 250y = 250
⇒ -210y + 250y = -222 + 250
⇒ 40y = 28
Transposing 40
⇒ y = 0.7
Putting the value of y in (v)
⇒ x = [74 − 70(0.7)] / 50
⇒ x = (74 – 49) / 50
⇒ x = 25/ 50 = 1/2
Therefore, x = 0.5
Therefore, x = 0.5 and y = 0.7
问题15. 1 /(7x)+ 1 /(6y)= 3和1 /(2x)– 1 /(3y)= 5
解决方案：
1/(7x) + 1/(6y) = 3………………………….. (i)
1/(2x) – 1/(3y) = 5……………………………. (ii)
Let 1/x = u and 1/y = v
u/7 + v/6 = 3 ……………………………. (iii)
u/2 – v/3 = 5 ………………………………. (iv)
Taking LCM as 42 in (iii) and 6 in (iv)
6u + 7v = 126………………………………… (v)
3u – 2v = 30…………………………………. (vi)
Multiplying (vi) by 2
6u – 4v = 60…………………. (vii)
Subtracting (vii) from (v)
⇒ 6u – 6u +7v +4v = 126 – 60
⇒ 11v = 66
Transposing 11
⇒ v = 66/11
⇒ v = 6
⇒ y = 1/v
⇒y = 1/6
Putting v in (vii)
⇒ 6u – 4(6) = 60
⇒ 6u = 60 + 24
⇒ 6u = 84
Transposing 6
⇒ u = 84/6
⇒ u = 14
⇒ x = 1/u = 1/14
Therefore, x=1/14 and y=1/6 respectively.
问题16.1 /(2x)+ 1 /(3y)= 2和1 /(3x)+ 1 /(2y)= 13/6
解决方案：
Let 1/x = u and 1/y = v
u/2 + v/3 = 2 ………………(i)
u/3 + v/2 = 13/6 ……………(ii)
Taking 6 as LCM in (i) and (ii)
3u + 2v = 12…………… (iii)
2u + 3v = 13……………. (iv)
From (iii)
⇒ 3u = 12 – 2v
⇒ u = (12 – 2v) / 3
Putting in (iv)
⇒ 2(12 – 2v) / 3 + 3v = 13
⇒ (24 – 4v) / 3 + 3v = 13
Taking 3 as LCM
⇒ 24 – 4v +9v = 39
⇒ 5v = 39 – 24
⇒ 5v = 15
⇒ v= 3
⇒ y = 1/v = 1/3
Putting the value of v in (iii)
⇒ 3u + 2(3) = 12
⇒ 3u + 6 = 12
⇒ 3u = 6
⇒ u = 2
⇒ x = 1/u = 1/2
Therefore, x = 1/2, y = 1/3
问题17.15 / u + 2 / v = 17和1 / u + 1 / v = 36/5
解决方案：
Let 1/u = x and 1/v = y
15x + 2y = 17 ………………………….. (i)
x + y = 36/5………………………. (ii)
From (i)
2y = 17 – 15x
⇒ y = (17 − 15x) / 2 …………………. (iii)
Substituting (iii) in equation (ii)
⇒ x + (17 − 15x) / 2 = 36/5
Taking 2 as LCM
⇒ 2x + 17 – 15x = (36 × 2)/ 5
⇒ -13x = 72/5 – 17
Taking 5 as LCM
⇒ -13x = (72 – 85) / 5
⇒ -13x = -13/5
⇒ x = 1/5
⇒ u = 1/x = 5
Putting x = 1/5 in (ii)
1/5 + y = 36/5
⇒ y = 35/5
⇒ y = 7
⇒ v = 1/y = 1/7
Therefore, u = 5 and v = 1/7
问题18. 3 / x – 1 / y = -9和2 / x + 3 / y = 5
解决方案：
Let 1/x = u and 1/y = v
3u – v = -9…………………..(i)
2u + 3v = 5 ……………………….(ii)
Multiplying (i) by 3
9u – 3v = -27 ………………………….. (iii)
2u + 3v = 5 ……………………………… (iv)
Adding equation (iii) and (iv)
9u + 2u – 3v + 3v = -27 + 5
⇒ 11u = -22
⇒ u = -2
Putting u = -2 in (iv)
2(-2) + 3v = 5
⇒ -4 + 3v = 5
⇒ 3v = 9
⇒ v = 3
Therefore, x = 1/u = −1/2, y = 1/v = 1/3
问题19. 2 / x + 5 / y = 1和60 / x + 40 / y = 19
解决方案：
Let 1/x = u and 1/y = v
2u + 5v = 1…………………..(i)
60u + 40v = 19 ……………………….(ii)
Multiplying equation by 8
16u + 40v = 8 ………………………….. (iii)
60u + 40v = 19 ……………………………… (iv)
Subtracting equation (iii) from (iv)
60u – 16u + 40v – 40v = 19 – 8
⇒ 44u = 11
⇒ u = 11/44
⇒ u = 1/4
Putting u = 1/4 in (iv)
60(1/4) + 40v = 19
⇒ 15 + 40v = 19
⇒ 40v = 4
⇒ v = 4/ 40 = 1/10
x = 1/u = 4
y = 1/v = 10
问题20. 1 /(5x)+ 1 /(6y)= 12和1 /(3x)– 3 /(7y)= 8
解决方案：
Let 1/x = u and 1/y = v
u/5 + v/6 = 12…………………..(i)
u/3 – 3v/7 = 8……………………….(ii)
Taking LCM for both equations
6u + 5v = 360………. (iii)
7u – 9v = 168……….. (iv)
Subtracting (iii) from (iv)
7u – 9v – (6u + 5v) = 168 – 360
⇒ u – 14v = -192
⇒ u = (14v – 192)………. (v)
Using (v) in equation (iii)
6(14v – 192) + 5v = 360
⇒ 84v -1152 + 5v = 360
⇒ 89v = 1512
⇒ v = 1512/89
⇒ y = 1/v = 89/1512
Now, substituting v in equation (v)
u = 14 x (1512/89) – 192
⇒ u = 21168/89 – 192
⇒ u = (21168 – 17088) / 89
⇒ u = 4080/89
⇒ x = 1/u = 89/ 4080
Therefore, x = 89/4080 and y = 89/ 1512
问题21. 4 / x + 3y = 14和3 / x – 4y = 23
解决方案：
Taking 1/x = u
4u + 3y = 14…………………….. (i)
3u – 4y = 23…………………….. (ii)
Adding (i) and (ii), we get
4u + 3y + 3u – 4y = 14 + 23
⇒ 7u – y = 37
⇒ y = 7u – 37……………………… (iii)
Putting (iii) in (i),
4u + 3(7u – 37) = 14
⇒ 4u + 21u – 111 = 14
⇒ 25u = 125
⇒ u = 5
⇒ x = 1/u = 1/5
Putting u= 5 in (iii)
y = 7(5) – 37
⇒ y = -2
Therefore, x = 1/5 and y = -2
问题22. 4 / x + 5y = 7和3 / x + 4y = 5
解决方案：
Taking 1/x = u
4u + 5y = 7…………………….. (i)
3u + 4y = 5…………………….. (ii)
Multiplying (i) by 4
⇒ 16u + 20y = 28 …….. (iii)
Multiplying (ii) by 5
⇒ 15u + 20y = 25 …….. (iv)
(iii) – (iv)
⇒ 16u – 15u + 20y – 20y = 28–25
⇒ u = 3
⇒ x = 1/u = 1/3
Putting u in (i)
⇒ 12 + 5y = 7
⇒ 5y = -5
⇒ y = -1
Therefore, x = 1/3 and y = -1
问题23. 2 / x + 3 / y = 13和5 / x – 4 / y = -2
解决方案：
Let 1/x = u and 1/y = v
2u + 3v = 13………………….. (i)
5u – 4v = -2 ………………………. (ii)
Multiplying (i) by 4
⇒ 8u + 12v = 52 ……….. (iii)
Multiplying (ii) by 3
⇒ 15u – 12v = -6 ……….. (iv)
Adding (iii) and (iv)
⇒ 15u + 8u + 12v – 12v = 52 – 6
⇒ 23u = 46
⇒ u = 46/23
⇒ u = 2
Putting u = 2 in (i)
⇒ 4 + 3v = 13
⇒ 3v = 9
⇒ v = 3
⇒ x = 1/u = 1/2
⇒ y = 1/v = 1/3
Therefore, x = 1/2 and y = 1/3
问题24. 2 /√x+ 3 /√y= 2和4 /√x– 9 /√y= -1
解决方案：
Let 1/x = u and 1/y = v
2u + 3v = 2………………….. (i)
4u – 9v = -1 ………………………. (ii)
Multiplying (i) by 3
⇒ 6u + 9v = 6 …….. (iii)
Adding (ii) and (iii)
6u + 9v + 4u – 9v = 6 – 1
⇒ 10u = 5
⇒ u = 1/2
Substituting u = 1/2 in (i)
2(1/2) + 3v = 2
⇒ 3v = 2 – 1
⇒ v = 1/3
1/√x = u
⇒ x = 1/u2
⇒ x = 1/(1/2)2 = 4
1/√y = v
⇒ y = 1/v2
⇒ y = 1/(1/3)2 = 9
Therefore, x = 4 and y = 9.
问题25.(x + y)/ xy = 2和(x – y)/ xy = 6
解决方案：
(x + y)/xy = 2
⇒ 1/y + 1/x = 2……. (i)
(x – y)/xy = 6
⇒ 1/y – 1/x = 6………(ii)
Let 1/x = u and 1/y = v
v + u = 2……. (iii)
v – u = 6……..(iv)
Adding (iii) and (iv)
2v = 8
⇒ v = 4
⇒ y = 1/v = 1/4
Substituting v = 4 in (iii)
4 + u = 2
⇒ u = -2
⇒ x = 1/u = -1/2
Therefore, x = -1/2 and y = 1/4
问题26. 2 / x + 3 / y = 9 / xy和4 / x + 9 / y = 21 / xy
解决方案：
Taking LCM as xy
(2y + 3x)/ xy = 9/xy
⇒ 3x + 2y = 9………. (i)
(4y + 9x)/ xy = 21/xy
⇒ 9x + 4y = 21………(ii)
Multiplying (i) by 2
⇒ 6x + 4y = 18 …….. (iii)
(ii) – (iii)
⇒ 9x – 6x + 4y – 4y = 21–18
⇒ 3x = 3
⇒ x = 1
Putting x = 1 in (i)
3(1) + 2y = 9
⇒ y = 6/2
⇒ y = 3
Therefore, x = 1 and y = 3