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Question Number 155372 by mathdanisur last updated on 29/Sep/21

Commented by Rasheed.Sindhi last updated on 30/Sep/21

What′s meant by Z.Q ?

What′smeantbyZ.Q?

Commented by mathdanisur last updated on 30/Sep/21

Dear Ser, Z-integers an Q-rationl

DearSer,Z−integersanQ−rationl

Commented by Rasheed.Sindhi last updated on 01/Oct/21

I asked about Z . Q Do you mean x∈Z, y∈Q ?

IaskedaboutZ.QDoyoumeanx∈Z,y∈Q?

Commented by mathdanisur last updated on 30/Sep/21

Dear Ser, yes

DearSer,yes

Answered by Rasheed.Sindhi last updated on 30/Sep/21

((x^2 +xy+y^2 )/(x+2y))=(7/5) 5x^2 +5xy+5y^2 −7x−14y=0 5y^2 +(5x−14)y+5x^2 −7x=0 y=((−(5x−14)±(√((5x−14)^2 −4(5)(5x^2 −7x))))/(10)) △=25x^2 −140x+196−100x^2 +140x =25x^2 +196−100x^2 =196−75x^2 For y being rational △ must be perfect square. 196−75x^2 is perfect square for x=0,±1 only,because x≥2 △ is −ve and y will be imaginary. x=0⇒y=((−(5(0)−14)±(√(196−75(0)^2 )))/(10)) =((14±14)/(10))=0,((14)/5) (0,0) is not acceptable,because it doesn′t satisfy the original equatkon. (x,y)=(0,((14)/5)) x=1,y=((−(5−14)±11)/(10))=((9±11)/(10))= y=2,−(1/5) (x,y)=(1,2)(1,−(1/5)) x=−1⇒y=((−(5(−1)−14)±11)/(10)) =((5+14±11)/(10))=3,(4/5) (x,y)=(−1,3),(−1,(4/5))

x2+xy+y2x+2y=755x2+5xy+5y2−7x−14y=05y2+(5x−14)y+5x2−7x=0y=−(5x−14)±(5x−14)2−4(5)(5x2−7x)10△=25x2−140x+196−100x2+140x=25x2+196−100x2=196−75x2Forybeingrational△mustbeperfectsquare.196−75x2isperfectsquareforx=0,±1only,becausex⩾2△is−veandywillbeimaginary.x=0⇒y=−(5(0)−14)±196−75(0)210=14±1410=0,145(0,0)isnotacceptable,becauseitdoesn′tsatisfytheoriginalequatkon.(x,y)=(0,145)x=1,y=−(5−14)±1110=9±1110=y=2,−15(x,y)=(1,2)(1,−15)x=−1⇒y=−(5(−1)−14)±1110=5+14±1110=3,45(x,y)=(−1,3),(−1,45)

Commented by mathdanisur last updated on 01/Oct/21

Very nice Ser, thank you

VeryniceSer,thankyou

Commented by mathdanisur last updated on 30/Sep/21

Thank you dear Ser but amswer: (x;y)=(-1;3),(-1;(4/5);),(0;((14)/5)),(1;2),(1;-(1/5))

ThankyoudearSerbutamswer:(x;y)=(−1;3),(−1;45;),(0;145),(1;2),(1;−15)

Commented by Rasheed.Sindhi last updated on 30/Sep/21

I′ve revised my answer and now it consist of all the answers.

I′verevisedmyanswerandnowitconsistofalltheanswers.

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