xy/x+y=1/2,yz/y+z=1/3,xz/x+z=1/4求3x+2y+5z

Question

xy/x+y=1/2,yz/y+z=1/3,xz/x+z=1/4
求3x+2y+5z

Answer 1

由题意得：x,y,z均不为0. 并且可得：
2xy =x+y (1)
3yz =y+z (2)
4xz = x+z (3)
(2):(3)得： 3y(x+z) = 4x(y+z) xy = 3yz - 4xz (4)
(1):(3)得： y(x+z) = 2z(x+y) xy = yz + 2xz (5)
由(4)(5)得：2yz = 6xz y = 3x (6)
把(6)代入(1)得：x = 2/3,∴y = 2
把y = 2代入(2)得：z = 2/5
把x = 2/3,y = 2,z = 2/5 代入3x + 2y + 5z 得
原式 = 2 + 4 + 2 = 8