Menu Close

repost-old-question-unanswer-Given-4x-2-1-4x-2-y-4y-2-1-4y-2-z-4z-2-1-4z-2-x-




Question Number 106794 by bobhans last updated on 07/Aug/20
repost old question unanswer  Given → { ((((4x^2 )/(1+4x^2 )) = y)),((((4y^2 )/(1+4y^2 )) = z)),((((4z^2 )/(1+4z^2 )) = x)) :}
repostoldquestionunanswerGiven{4x21+4x2=y4y21+4y2=z4z21+4z2=x
Answered by thearith last updated on 07/Aug/20
Answered by john santu last updated on 07/Aug/20
   ^(@JS@)    { ((1+(1/(4x^2 )) = (1/y))),((1+(1/(4y^2 )) = (1/z))),((1+(1/(4z^2 )) = (1/x))) :}  ⇒(1/(4x^2 ))+(1/(4y^2 ))+(1/(4z^2 ))+3 = (1/x)+(1/y)+(1/z)  ⇒((1/(4x^2 ))−(1/x)+1)+((1/(4y^2 ))−(1/y)+1)+((1/(4z^2 ))−(1/z)+1)=0  (((4x^2 −4x+1)/(4x^2 )))+(((4y^2 −4y+1)/(4y^2 )))+(((4z^2 −4z+1)/(4z^2 )))=0  (((2x−1)/(2x)))^2 +(((2y−1)/(2y)))^2 +(((2z−1)/(2z)))^2 =0   { ((x = (1/2))),((y=(1/2))),((z=(1/2))) :}.
@JS@{1+14x2=1y1+14y2=1z1+14z2=1x14x2+14y2+14z2+3=1x+1y+1z(14x21x+1)+(14y21y+1)+(14z21z+1)=0(4x24x+14x2)+(4y24y+14y2)+(4z24z+14z2)=0(2x12x)2+(2y12y)2+(2z12z)2=0{x=12y=12z=12.
Commented by bemath last updated on 07/Aug/20
thank you
thankyou
Commented by bobhans last updated on 07/Aug/20
and the other solution  { ((x=0)),((y=0)),((z=0)) :}
andtheothersolution{x=0y=0z=0
Answered by 1549442205PVT last updated on 07/Aug/20
WLOG suppose that x≥y≥z≥0.Then  from the hypothesis we have  ((4z^2 )/(1+4z^2 ))≥((4x^2 )/(1+4x^2 )) ≥((4y^2 )/(1+4y^2 ))     ⇒((4z^2 )/(1+4z^2 ))≥((4x^2 )/(1+4x^2 ))   ⇔4z^2 (1+4x^2 )≥4x^2 (1+4z^2 )⇔z^2 ≥x^2   ⇒x=y=z.Replace into (1) we get  4x^2 =x(1+4x^2 )⇔4x^3 −4x^2 +x=0  ⇔x(4x^2 −4x+1)=0⇔x(2x−1)^2 =0  ⇔x∈{0;(1/2)} Hence,the given system  has two solutions as  {(x,y,z)∈{(0,0,0);((1/2);(1/2),(1/2))}
WLOGsupposethatxyz0.Thenfromthehypothesiswehave4z21+4z24x21+4x24y21+4y24z21+4z24x21+4x24z2(1+4x2)4x2(1+4z2)z2x2x=y=z.Replaceinto(1)weget4x2=x(1+4x2)4x34x2+x=0x(4x24x+1)=0x(2x1)2=0x{0;12}Hence,thegivensystemhastwosolutionsas{(x,y,z){(0,0,0);(12;12,12)}
Commented by bemath last updated on 07/Aug/20
sir what is WLOG
sirwhatisWLOG
Commented by 1549442205PVT last updated on 07/Aug/20
it is the brief writting of phrase  “without loss the generality”is used  in the math. proof
itisthebriefwrittingofphrasewithoutlossthegeneralityisusedinthemath.proof

Leave a Reply

Your email address will not be published. Required fields are marked *