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February 7th, 2018, 08:50 PM  #1 
Banned Camp Joined: Apr 2017 From: durban Posts: 22 Thanks: 0 Math Focus: Algebra  Simultaneous equations
i have look into the solving of the equation just want to confirm if my calculation is right. Q: X + Y + Z = 1 X^2 + Y^2 + Z^2 = 35 X^3 + Y^3 + Z^3 = 97 FIND THE VALUE OF X, Y AND Z Last edited by greg1313; February 8th, 2018 at 03:47 PM. 
February 7th, 2018, 08:56 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,758 Thanks: 900 
just looking at the second equation I see that $35 = 25 + 9 + 1$ making that work for the first equation we'd get $1 = 5 + (3) + (1)$ now let's check via the third equation $(5)^3 + (3)^3 + (1)^3 = 125  27  1 = 97$ hurrah $x = 5, ~y=3,~z=1$ or since the system is completely symmetric any permutation of those equalities. 
February 8th, 2018, 12:40 AM  #3  
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84  Quote:
 
February 8th, 2018, 01:53 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,436 Thanks: 562 
Is there an underlying condition that only integer solutions be considered? If so, you are done.

February 8th, 2018, 02:04 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 1,758 Thanks: 900 
this is the basic algebra forum people. They aren't looking for real solutions of a nonlinear system of equations. 
February 8th, 2018, 07:28 PM  #6  
Banned Camp Joined: Apr 2017 From: durban Posts: 22 Thanks: 0 Math Focus: Algebra  Quote:
x = 1 y  z eqn.1 substitute eqn. 1 for eqn. 2 x^2 + Y^2 + z^2 = 35 (1  y  z)^2 + Y^2 + z^2 = 35  
February 8th, 2018, 11:54 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,574 Thanks: 1485 
(1  Y  Z)² + Y² + z² = 35 implies (Y + Z  2)² + 2(Y + Z  2)  16 = (Y  1)(Z  1). From X³ + Y³ + Z³ = 97, one gets (1  Y  Z)³ + Y³ + Z³ = 97, so 3Y² + 3Z²  3Y²Z  3YZ² + 6YZ  3Y  3Z = 96, which implies (Y  1)(Z  1)(Y + Z) = 32. Hence ((Y + Z  2)² + 2(Y + Z  2)  16)(Y + Z) = 32, which implies Y³ + Z³ + 3Y²Z + 3YZ²  2Y²  2Z²  4YZ  16Y  16Z + 32 = 0, which factorizes as (Y + Z + 4)(Y + Z  4)(Y + Z  2) = 0. Hence Y + Z = ±4 or 2. If Y + Z = 2, X = 1 and (Y  1)(Z  1) = 16. From there, it's easy to show that (Y, Z) = (3, 5) or (5, 3). I'll leave you to finish from there. 

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equations, simultaneous, sinmultaneous 
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