標題:
Maths question about ratio
發問:
If (2x - 3y)/(y + 3z) = (y – z)/(x – z) = (x+ 3z)/(2y – 3x), prove that each of these ratios is equal to x/y. Hence, showthat either x = y or x + y= z.
最佳解答:
Let (2x - 3y)/(y + 3z) = (y – z)/(x – z) = (x + 3z)/(2y – 3x) = k then 2x - 3y = k(y + 3z) ... (1) y - z = k(x - z) .......... (2) x + 3z = k(2y - 3x) ... (3)(1) + (2)*3 + (3) : 2x - 3y + 3y - 3z + x + 3z = k(y + 3z + 3x - 3z + 2y - 3x) 3x = k(3y) k = x / y Hence , by (2) , y - z = (x / y) (x - z) y2 - yz = x2 - xz x2 - y2 - z(x - y) = 0 (x - y) (x + y - z) = 0 x = y or x + y = z 2013-05-09 15:57:02 補充： If k = 0 , 2x - 3y = y - z = x + 3z = 0 ? x = y = z = 0 then 0/0 = k , so k ≠ 0. 2013-05-09 15:58:37 補充： If k = 0 , 2x - 3y = y - z = x + 3z = 0 ? x = y = z = 0 then 0/0 = k , so k ≠ 0. 2013-05-09 16:08:34 補充： If y = 0 , then x = 0 since k ≠ 0 , but k = 0/(y + 3z) = 0 (contradiction) , so y ≠ 0. 2013-05-09 16:08:55 補充： If y = 0 , then x = 0 since k ≠ 0 , but k = 0/(y + 3z) = 0 (contradiction) , so y ≠ 0.
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