一題數學題

x + y + z = 5 ...... (1)
x² + y² + z² = 15 ...... (2)
xy = z² ...... (3)

(1)² - (2):
(x + y + z)² - (x² + y² + z²) = 5² - 15
x² + y² + z² + 2xy + 2yz + 2zx - x² - y² - z² = 10
2xy + 2yz + 2zx = 10
xy + yz + zx = 5 ...... (4)

把 (3) 代入 (4) 中：
z² + yz + zx = 5
zx + yz + z² = 5
z(x + y + z) = 5 ...... (5)

(5)/(1):
z = 1

(3):
xy = z²
xyz = z³ ...... (6)

把 z = 1 代入 (6) 中：
xyz = 1 ...... (7)

1/x + 1/y + 1/z = yz/xyz + zx/xyz + xy/xyz
1/x + 1/y + 1/z = (yz + zx + xy)/xyz
1/x + 1/y + 1/z = (xy + yz + zx)/xyz ...... (8)

把 (4) 及 (7) 代入 (8) 中：
1/x + 1/y + 1/z = (5)/(1)
1/x + 1/y + 1/z = 5

(1)x+y+z=5=>z=5-x-y代入(2),(3)
(2)x^2+y^2+z^2=15
(3)xy=z^2
得
(2)’ x^2+ xy-5x+y^2-5y+5 = 0
(3)’ x^2+xy-10x+y^2-10y+25=0
(2)’- (3)’
得
(4)x+ y-4=0代入 (1) 得z=1代入(3) 得xy=1

由x+ y-4=0, xy=1聯立得:x = 2-√3, y = 2+√3, z=1
x = 2+√3, y = 2-√3 ,z=1

=>(1/x)+(1/y)+(1/z)=5