要求 1995(x+y) + 6xy - (17/2)(a+b),
還是 1995(x+y) + 6xy - 17/[2(a+b)]?
解:
ax + by = 7 ...... (1)
ax² + by² = 49 ...... (2)
ax³ + by³ = 133 ...... (3)
ax^4 + by^4 = 406 ...... (4)
(1)*xy + (3) 得
ax²y + bxy² + ax³ + by³ = 7xy + 133
ax²(x+y) + by²(x+y) = 7xy + 133
(x+y)(ax² + by²) = 7xy + 133
(x+y)(49) = 7xy + 133 ﹝by (2)﹞
7(x+y) = xy + 19 ...... (5)
(2)*xy + (4) 得
ax³y + bxy³ + ax^4 + by^4 = 49xy + 406
ax³(x+y) + by³(x+y) = 49xy + 406
(x+y)(ax³ + by³) = 49xy + 406
(x+y)(133) = 49xy + 406 ﹝by (3)﹞
19(x+y) = 7xy + 58 ...... (6)
(5), (6) 可解得﹝請自行練習﹞
x+y = 5/2 ...... (7)
xy = -3/2 ...... (8)
(7), (8) 可解得﹝請自行練習﹞
(x,y) = (3,-1/2) or (-1/2,3)
將 (x,y) = (3,-1/2) 分別代入 (1), (2) 可解得 (a,b) = (5,16)。﹝請自行練習﹞
同理,基於 (1), (2) 的 x, y 及 a, b 對稱性,
若將 (x,y) = (-1/2,3) 分別代入 (1), (2) 將得到 (a,b) = (16,5)。
故 a+b = 21 ...... (9)
將 (7), (8), (9) 代入所求即為答案。﹝請自行求解﹞ ■
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