The number of real solutions of the system of equations $$x=2z21+z2$$,$$y=2x21+x2$$,$$z=2y21+y2$$ is:

Question

The number of  real solutions of the system of equations $$x=2z21+z2$$,$$y=2x21+x2$$,$$z=2y21+y2$$      is:

Infinity Learn · Accepted Answer

The  given system of equations are :   $$x=2z21+z2$$              ……$$($$ i $$)$$ $$y=2x21+x2$$              ……$$($$ ii $$)$$ $$z=2y21+y2$$              ……$$($$ iii $$)Now$$,  $$x=2z1+z2$$.z                $$=2z1+z2.2y21+y2$$                 [from  Eq.(iii)]          $$=2z(1+z2).2y(1+y2)$$.y               $$=2z1+z2.2y1+y2.2x21+x2$$          [from Eq. (ii)]∴      $$1=(2z1+z2)(2y1+y2)(2x1+x2)$$     Let  $$x=$$$$tan$$  α,$$y=$$$$tan$$β      and $$z=$$$$tan$$γ           $$1=(2tan$$γ$$1+tan2$$γ$$)(2tan$$β$$1+tan2$$β$$)(2tan$$α$$1+tan2$$α$$)$$           ∴      $$1=sin2$$α$$sin2$$β$$sin2$$γ              $$($$∵$$sin2$$α$$=2tan$$α$$1+tan2$$α$$)$$     It  is possible only when             $$sin2$$α$$=sin2$$β$$=sin2$$γ$$=1$$     ∴        α$$=$$β$$=$$γ$$=450$$     ∴        $$x=y=z=1$$     ∴       Solution of the  given equation is (1$$,$$1$$,$$1)     .

The number of real solutions of the system of equations x=2z21+z2,y=2x21+x2,z=2y21+y2 is:

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