數學的疑難-解方程組兩題－香港美食2017

標題:

數學的疑難-解方程組兩題

發問:

1. x + y = axy -------(1) y + z = byz -------(2) z + x = czx -------(3) 2. x ( y + z - x ) = 30 - 2x2 -------(1) y ( x - y + z ) = 52 - 2y2 -------(2) z ( x + y - z ) = 78 - 2z2 -------(3)

最佳解答:

1)x + y = axy -------(1) y + z = byz -------(2) z + x = czx -------(3)當 x , y , z ≠ 0 :1/y + 1/x = a .....(1) 1/z + 1/y = b .....(2) 1/x + 1/z = c .....(3)((1) + (2) + (3)) / 2 :1/x + 1/y + 1/z = (a + b + c)/2 .....(4)(4) - (1) :1/z = (a + b + c)/2 - a z = 2 / (b + c - a)(4) - (2) :1/x = (a + b + c)/2 - b x = 2 / (a + c - b)(4) - (3) :1/y = (a + b + c)/2 - c y = 2 / (a + b - c)當 x , y , z 其一為0 , x = y = z = 0 2)x ( y + z - x ) = 30 - 2x2 -------(1) y ( x - y + z ) = 52 - 2y2 -------(2) z ( x + y - z ) = 78 - 2z2 -------(3)==>x (x + y + z) = 30 .....(1) y (x + y + z) = 52 .....(2) z (x + y + z) = 78 .....(3)(1) + (2) + (3) :(x + y + z)2 = 160 x + y + z = √160 或 - √160 , 代回 (1) , (2) , (3) :x = ± 30 / √160 y = ± 52 / √160 z = ± 78 / √160

其他解答: