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Is it possible to solve an exact solution for this system? WolframAlpha has an exact solution but I can't find a way to arrive at it myself.

$$x = 1 + \frac{\sqrt{4-y^2} - 1}{2}$$

$$y = 1 + \frac{\sqrt{4-x^2} - 1}{2}$$

Blue
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Jeff L
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  • Of course, but once I've inserted the equation for $y$ into the equation for $x$ I can't figure out how to simplify it much further. – Jeff L Mar 27 '22 at 02:53
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    First of all i would like to see your work on that. Furthermore, if eliminating x gives the above product that you commented, then you just have to find the roots and replace them in order to find $x$ also. – Giorgos Kosmas Mar 27 '22 at 02:57

1 Answers1

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Those equations can be written as $(2x-1)^2=4-y^2, (2y-1)^2=4-x^2$ after some trivial arithmetic manipulations. Substracting the first from the second we have $$4x^2-4x+y^2=4y^2-4y+x^2$$ or equivalently $$(x-y)(3x-3y-4)=0,$$ which implies that $$(x=y)\lor (x=y+4/3).$$ I hope you can handle the rest.

P.S. I skipped some steps but I guess you can recover them (though this is a good exercise).

RFZ
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