maths(解方程 二 : 代數方法)

16. 設她原來買入玻璃杯共x隻
設售價為$y
原來買入價 = $(y - 3)
總買入價= 48 = x(y - 3) ... (1)
總售價 = y(x - 2) = 48 + 22 ... (2)
(1) => 48 = xy - 3x ...(3)
(2) => xy - 2y = 70 ... (4)
(3) + (4) => 48 + xy - 2y = xy - 3x + 70
48 - 2y = -3x + 70
2y = 3x - 22
y = 1.5x - 11 ...(3)
代入(1), 48 = x(1.5x - 14)
48 = 1.5x^2 - 14x
1.5x^2 - 14x - 48 = 0
3x^2 - 28x - 96 = 0
(x - 12)(3x + 8) = 0
x = 12 或 x = -8/3 (不合)
a) 她原來買入玻璃杯共12隻
b) 代入(3), y = 1.5(12) - 11 = 7
每隻玻璃杯的售價是$7
17. 設每個玻璃相架的成本為$x
設共買入N個
總成本 = $Nx = $3600 => N = 3600/x ... (1)
第一個星期總售價 = 60(x + 18)
第二個星期總售價 = (N - 60 - 12)(x - 6)
總利潤 = 60(x + 18) + (N - 60 - 12)(x - 6) - 3600 = 480 ... (2)
代(1)入(2) => 60x + 1080 + (3600/x - 72)(x - 6) - 4080 = 0
60x + 3600 - 72x - 21600/x + 432 - 3000 = 0
-12x + 1032 - 21600/x = 0
-12x^2 + 1032x - 21600 = 0
x^2 - 86x + 1800 = 0
(x - 50)(x - 36) = 0
x = 50 或 x = 36
x = 50 => N = 72 (不合)
x = 36 => N = 100
每個玻璃相架的成本為 $36
18. (a) 計算機每部買價 = 2400/x
售價 = 2400/x + 50
總售價 = (2400/x + 50)(x - 1) = 3000
2400 + 50x - 2400/x - 50 = 3000
50x - 650 - 2400/x = 0
50x^2 - 650x - 2400 = 0
x^2 - 13x - 48 = 0
(x - 16)(x + 3) = 0
x = 16 或x = -3 (不合)
志強共購入了16部計算機
b) 計算機每部買價 = 2400/16 = $150
他以150 + 50 = $200售出每部計算機