Prove that for all real xx and yy, if x(x+y−1)+y(y−1)=0x(x+y-1)+y(y-1)=0 then 0≤x+y≤430\le x+y\le\frac43.

Any help will be appreciated, this problem did really frustrate me, I know that I’m not allowed to use the ellipse equation because the problem is supposed to be solved using mathematical logic.

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