System of equations problem

Question

This two equations. How we can solve them? $$ \begin{cases} \displaystyle\frac{1}{\sqrt{x+1} +1} +\frac{1}{\sqrt{y+1} +1} =\frac{2}{3} \\[6px] \displaystyle\sqrt{\frac{1}{x^{2}} +\frac{1}{y\vphantom{^2}} } +\sqrt{\frac{1}{y^{2}} +\frac{1}{x} } -\frac{2}{\sqrt{xy} } =\frac{2}{3} \end{cases}$$ I tried using inequalities and other ways but I couldn't solve it. My question is: how we can solve this system of equations?

Answer 1

Eliminating square roots leads to $(x,y)=(3,3)$ but also many spurious solutions from combinations of signs of the square roots:

In M2

R=QQ[x,y,s,t]
F=frac R
ideal(1/(s+1)+1/(t+1)-2/3,s^2-x-1,t^2-y-1) -- ideal((-2*s*t+s+t+4)/(3*s*t+3*s+3*t+3),s^2-x-1,t^2-y-1)
R=QQ[s,t,x,y,MonomialOrder=>Eliminate 2]
I1=ideal(-2*s*t+s+t+4,s^2-x-1,t^2-y-1)
gens gb I1 -- 16*x^2*y^2+24*x^2*y+24*x*y^2+9*x^2-162*x*y+9*y^2-216*x-216*y

R=QQ[x,y,u,v,w]
F=frac R
ideal(u+v-2/w-2/3,1/x^2+1/y-u^2,1/y^2+1/x-v^2,x*y-w^2) -- ideal((3*u*w+3*v*w-2*w-6)/(3*w),(-x^2*y*u^2+x^2+y)/(x^2*y),(-x*y^2*v^2+y^2+x)/(x*y^2),x*y-w^2)
R=QQ[u,v,w,x,y,MonomialOrder=>Eliminate 3]
I2=ideal(3*u*w+3*v*w-2*w-6,-x^2*y*u^2+x^2+y,-x*y^2*v^2+y^2+x,x*y-w^2)
gens gb I2 -- 256*x^8*y^8-2304*x^8*y^7-2304*x^7*y^8+5472*x^8*y^6-4032*x^7*y^7+5472*x^6*y^8-6480*x^8*y^5+47952*x^7*y^6+47952*x^6*y^7-6480*x^5*y^8+26001*x^8*y^4+64476*x^7*y^5-367578*x^6*y^6+64476*x^5*y^7+26001*x^4*y^8-14580*x^8*y^3-527796*x^7*y^4+355752*x^6*y^5+355752*x^5*y^6-527796*x^4*y^7-14580*x^3*y^8+27702*x^8*y^2+72900*x^7*y^3+998730*x^6*y^4-1265544*x^5*y^5+998730*x^4*y^6+72900*x^3*y^7+27702*x^2*y^8-26244*x^8*y+131220*x^7*y^2+183708*x^6*y^3-288684*x^5*y^4-288684*x^4*y^5+183708*x^3*y^6+131220*x^2*y^7-26244*x*y^8+6561*x^8-104976*x^7*y+603612*x^6*y^2-1574640*x^5*y^3+2138886*x^4*y^4-1574640*x^3*y^5+603612*x^2*y^6-104976*x*y^7+6561*y^8

S=QQ[x,y]
J=ideal(16*x^2*y^2+24*x^2*y+24*x*y^2+9*x^2-162*x*y+9*y^2-216*x-216*y,256*x^8*y^8-2304*x^8*y^7-2304*x^7*y^8+5472*x^8*y^6-4032*x^7*y^7+5472*x^6*y^8-6480*x^8*y^5+47952*x^7*y^6+47952*x^6*y^7-6480*x^5*y^8+26001*x^8*y^4+64476*x^7*y^5-367578*x^6*y^6+64476*x^5*y^7+26001*x^4*y^8-14580*x^8*y^3-527796*x^7*y^4+355752*x^6*y^5+355752*x^5*y^6-527796*x^4*y^7-14580*x^3*y^8+27702*x^8*y^2+72900*x^7*y^3+998730*x^6*y^4-1265544*x^5*y^5+998730*x^4*y^6+72900*x^3*y^7+27702*x^2*y^8-26244*x^8*y+131220*x^7*y^2+183708*x^6*y^3-288684*x^5*y^4-288684*x^4*y^5+183708*x^3*y^6+131220*x^2*y^7-26244*x*y^8+6561*x^8-104976*x^7*y+603612*x^6*y^2-1574640*x^5*y^3+2138886*x^4*y^4-1574640*x^3*y^5+603612*x^2*y^6-104976*x*y^7+6561*y^8)
primaryDecomposition J -- {ideal(40*x^2*y+40*x*y^2+39*x^2+578*x*y+39*y^2+600*x+600*y,360*x^3+360*y^3-5553*x^2+225394*x*y-5553*y^2+283800*x+283800*y,7200*y^4+4715960*x*y^2-104040*y^3-13041*x^2+12842418*x*y+5770959*y^2+8501400*x+8501400*y), ideal(7*x*y+3*x+3*y-81,7*x^2+7*y^2-90*x-90*y+414), ideal(x*y+3*x+3*y+9,x^2+y^2+6*x+6*y+18),ideal(16*x^2*y^2+24*x^2*y+24*x*y^2+9*x^2-162*x*y+9*y^2-216*x-216*y,320*x^4*y+320*x*y^4+168*x^4-480*x^3*y-480*x*y^3+168*y^4-4932*x^3-8100*x^2*y-8100*x*y^2-4932*y^3+21105*x^2+65806*x*y+21105*y^2-47256*x-47256*y-73728,2880*x^5+2880*y^5-73512*x^4-2895200*x^3*y-2895200*x*y^3-73512*y^4-74292*x^3+32471260*x^2*y+32471260*x*y^2-74292*y^3+22251495*x^2-103951934*x*y+22251495*y^2-138095016*x-138095016*y+41951232)}

In maxima CAS

solve([7*x*y+3*x+3*y-81,7*x^2+7*y^2-90*x-90*y+414],[x,y]); 
# [[x = 3,y = 3]]
solve([40*x^2*y+40*x*y^2+39*x^2+578*x*y+39*y^2+600*x+600*y,360*x^3+360*y^3-5553*x^2+225394*x*y-5553*y^2+283800*x+283800*y,7200*y^4+4715960*x*y^2-104040*y^3-13041*x^2+12842418*x*y+5770959*y^2+8501400*x+8501400*y],[x,y]);
# [[x = 0,y = 0]]
solve([x*y+3*x+3*y+9,x^2+y^2+6*x+6*y+18],[x,y]);
# [[x = -3,y = -3]]
solve([16*x^2*y^2+24*x^2*y+24*x*y^2+9*x^2-162*x*y+9*y^2-216*x-216*y,320*x^4*y+320*x*y^4+168*x^4-480*x^3*y-480*x*y^3+168*y^4-4932*x^3-8100*x^2*y-8100*x*y^2-4932*y^3+21105*x^2+65806*x*y+21105*y^2-47256*x-47256*y-73728,2880*x^5+2880*y^5-73512*x^4-2895200*x^3*y-2895200*x*y^3-73512*y^4-74292*x^3+32471260*x^2*y+32471260*x*y^2-74292*y^3+22251495*x^2-103951934*x*y+22251495*y^2-138095016*x-138095016*y+41951232],[x,y]);
# [[x = 15.35469448584203,y = 0.2879050925925926],
    [x = (-0.9055397242585678*%i)-0.4046277369000439,
     y = 0.9055397242585677*%i-0.4046277369000441],
    [x = 0.9055397242585678*%i-0.404627736900044,
     y = (-0.9055397242585677*%i)-0.4046277369000441],
    [x = (-18.67256559242221*%i)-6.003500862860708,
     y = 0.7332666307739744*%i-0.5919296465532026],
    [x = 18.67256559242221*%i-6.00350086286071,
     y = (-0.7332666307739744*%i)-0.5919296465532026],
    [x = 7.709339774557166,y = 1.010534236267871],
    [x = 2.997085916999494-3.654663624319716*%i,
     y = 0.1289410091061662*%i-0.9874526543022433],
    [x = 3.654663624319717*%i+2.997085916999493,
     y = (-0.1289410091061662*%i)-0.9874526543022433],
    [x = 6.303107861060329,y = 1.315268368942157],
    [x = (-0.1289410091061663*%i)-0.9874526543022434,
     y = 3.654663624319717*%i+2.997085916999493],
    [x = 0.1289410091061664*%i-0.9874526543022435,
     y = 2.997085916999493-3.654663624319717*%i],
    [x = 1.315268368942157,y = 6.303107861060329],
    [x = 1.010534236267871,y = 7.709339774557166],
    [x = 0.2879050925925926,y = 15.35469448584203],
    [x = (-0.7332666307739742*%i)-0.5919296465532023,
     y = 18.67256559242221*%i-6.003500862860712],
    [x = 0.7332666307739742*%i-0.5919296465532023,
     y = (-18.67256559242221*%i)-6.003500862860712]]