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Question-159469




Question Number 159469 by 0731619 last updated on 17/Nov/21
Commented by cortano last updated on 17/Nov/21
 (√x) = u and (√y) = v     { ((u^3 +v^3  = 35)),((u^2  v+ uv^2  = 30)) :}    { ((u^3 +v^3  = 35⇒(u+v)^3 −3uv(u+v)=35)),((uv (u+v)= 30)) :}  ⇒(u+v)^3  = 125 ⇒u+v=5  ⇒uv = 6 ⇒u(5−u)=6  ⇒u^2 −5u+6=0 ⇒ { ((u=3 ; v=2)),((u=2 ; v=3)) :}  ⇒ { ((x=9 ; y=4)),((x=4 ; y=9)) :}
x=uandy=v{u3+v3=35u2v+uv2=30{u3+v3=35(u+v)33uv(u+v)=35uv(u+v)=30(u+v)3=125u+v=5uv=6u(5u)=6u25u+6=0{u=3;v=2u=2;v=3{x=9;y=4x=4;y=9
Answered by FongXD last updated on 17/Nov/21
 { (((√x^3 )+(√y^3 )=35   (1))),(((√(x^2 y))+(√(xy^2 ))=30   (2))) :}  • take (1)+3(2):  ⇔ (√x^3 )+3(√(x^2 y))+3(√(xy^2 ))+(√y^3 )=125  ⇔ ((√x)+(√y))^3 =5^3 , ⇒ (√x)+(√y)=5   (3)  square both sides, ⇔ x+y+2(√(xy))=25  ⇒ x+y−(√(xy))=25−3(√(xy))  (1): (√x^3 )+(√y^3 )=35  ⇔ ((√x)+(√y))(x−(√(xy))+y)=35  ⇔ 5(25−3(√(xy)))=35  ⇔ (√(xy))=6, ⇒ xy=36     if you want to find the values of x and y which  satisfy the system of Eq. above, just substitute (√x)=(6/( (√y))) into (3).
{x3+y3=35(1)x2y+xy2=30(2)take(1)+3(2):x3+3x2y+3xy2+y3=125(x+y)3=53,x+y=5(3)squarebothsides,x+y+2xy=25x+yxy=253xy(1):x3+y3=35(x+y)(xxy+y)=355(253xy)=35xy=6,xy=36ifyouwanttofindthevaluesofxandywhichsatisfythesystemofEq.above,justsubstitutex=6yinto(3).
Commented by Rasheed.Sindhi last updated on 17/Nov/21
Nice approach!
Niceapproach!

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