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Simultaneous Settlement V (Posted on 2016-10-24) Difficulty: 2 of 5
Consider the following system of simultaneous equations:

X+Y = 1 and, X3+Y3 = 19

Without solving for X and Y, determine X9+Y9

No Solution Yet Submitted by K Sengupta    
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solution | Comment 1 of 2
(i)
x^3+y^3=19
(x+y)(x^2-xy+y^2)=19
x^2-xy+y^2=19

(ii)
(x+y)^3=1
x^3+3x^2y+3xy^2+y^3=1
x^3+3xy(x+y)+y^3=1
(x^3+y^3)+3xy(x+y)=1
19+3xy=1
3xy=-18
xy=-6

(iii)
(x^3+y^3)^2=19^2
x^6+2x^3y^3+y^6=361
x^6+y^6=361-2*(-6)^3
x^6+y^6=793

(iv)
x^9+y^9=(x+y)(x^2-xy+y^2)(x^6-x^3y^3+y^6)
x^9+y^9=1*19*(793-(-6)^3)
x^9+y^9=1*19*1009
x^9+y^9=19171

Direct verification:
x^3+(1-x)^3=19
3x^2-3x+1=19
3x^2-3x-18=0
x^2-x-6=0
(x-3)(x+2)=0
x=3 or x=-2
x=3 -> y=-2
x=-2 -> y=3
3^9+(-2)^9=19171

Edited on October 24, 2016, 8:20 am
  Posted by Daniel on 2016-10-24 08:15:16

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