中三數學題一條

Solution:

x+y=10

y=10-x

x^3+(10-x)^3=400

[x+(10-x)][x^2-x(10-x)+(10-x)^2]=400

10[x^2-10x+x^2+100-20x+x^2]=400

3x^2-30x+100=40

3x^2-30x+60=0

x^2-10x+20=0

x=7.24 or 2.76

y=2.76 or 7.24

the solution is

{ x=7.24,y=2.76}
{ x=2.76,y=7.24}

(correct to 3 significant figures)

我用Maxima黎計，答案如下
x=5-sqrt(5), y=sqrt(5)+5　或
x=sqrt(5)+5, y=5-sqrt(5)

sqrt是平方

用自己僅有的知識黎計：
x+y=10 ----i
x^3+y^3=400 ----ii
從i,x=10-y ----iii
從ii,(x-y)(x^2-xy+y^2)=400 ----iv
把iii代入iv,可得
(10-y+y)[(10-y)^2-(10-y)y+y^2]=400
10[(10-y)^2-10y+y^2+y^2]=400
10^2-20y+y^2-10+y^2+y^2=40
100-20y-10y+3(y^2)=40
-30y+3y^2= -60
3y(-10+y)= -60
-10+y= -60/3y
-10+y= -20/y
-10y+y^2=20
y(-10+y)=20
y=20/(-10+y)

把y=20/(-10+y)代入iii,可得
x=10-20/(-10+y)

計到走火入魔,不過答案係上面

2007-05-15 18:24:42 補充：
x=5 － 平方5,y=平方5 ＋ 5或x=平方5 5,y=5 - 平方5小數：x=2.763932......y=7.2360......或x=2.7639......y=7.2360......

x+y=10-----------------------(1)
x^3+y^3=400----------------(2)


x+y=10
x=10- y

代入
x^3+y^3=400

(10-y)^3+y^3=400

(10-y)(10-y)(10-y)+y^3=400

(100-20y + y^2 ))(10-y)+y^3=400


1000-100y-20y+20y^2 +10 y^2 -y^3+y^3 =400

1000-120y +30y^2 =400

600-120y +30y^2=0

20 - 4y + y^2 =0




呢條數係冇解ga....

x+y = 10 ---> y = 10-x

x^3 + y^3 = 400 ---> x^3 + (10-x)^3 = 400

x^3 + (x^2 - 20x +100)(10-x) = 400

x^3 + (-x^3 + 10x^2 + 20x^2 - 200x - 100x + 1000) = 400

拆()
30x^2 - 300x = -600 ---> 30x^2 - 300x + 600 = 0

將之除開30
--> x^2 - 10x + 20 = 0

計到x 出黎

再將x 代入第一條式入面, 就會計埋y 出黎

x+y=10 -------------------(i)
x^3+y^3=400 -----------(ii)

From(i), we have x=10-y -----(iii)

Sub (iii) into (ii),

(10-y)^3+y^3=400

3y^2-30y+60=0

y= 2.76 or 7.24

x=2.76 or 7.24