x^2+y=31 y^2+x=41

$$x^{2} + y = 31$$

2
y + x = 41

$$x + y^{2} = 41$$

$$x_{1} = 5$$
=
$$5$$
=

5

$$y_{1} = 6$$
=
$$6$$
=

6

$$x_{2} = 41 – left(-2 + left(- frac{1}{2} – frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}} + frac{58}{3 left(- frac{1}{2} – frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}}right)^{2}$$
=
$$frac{1}{144 left(1 + sqrt{3} iright)^{2} left(1503 + sqrt{82335} iright)^{frac{2}{3}}} left(- left(464 sqrt[3]{18} + left(1 + sqrt{3} iright) left(24 + sqrt[3]{12} left(1 + sqrt{3} iright) sqrt[3]{1503 + sqrt{82335} i}right) sqrt[3]{1503 + sqrt{82335} i}right)^{2} + 5904 left(1 + sqrt{3} iright)^{2} left(1503 + sqrt{82335} iright)^{frac{2}{3}}right)$$
=

6.07533614598892 – 1.49182035761475e-28*i

$$y_{2} = -2 + left(- frac{1}{2} – frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}} + frac{58}{3 left(- frac{1}{2} – frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}}$$
=
$$- frac{1}{left(1 + sqrt{3} iright) sqrt[3]{1503 + sqrt{82335} i}} left(frac{116 sqrt[3]{18}}{3} + frac{1}{12} left(1 + sqrt{3} iright) left(24 + left(1 + sqrt{3} iright) sqrt[3]{18036 + 12 sqrt{82335} i}right) sqrt[3]{1503 + sqrt{82335} i}right)$$
=

-5.90970928675947 + 0.e-22*i

$$x_{3} = 41 – left(-2 + frac{58}{3 left(- frac{1}{2} + frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}} + left(- frac{1}{2} + frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}right)^{2}$$
=
$$frac{1}{144 left(1 – sqrt{3} iright)^{2} left(1503 + sqrt{82335} iright)^{frac{2}{3}}} left(5904 left(1 – sqrt{3} iright)^{2} left(1503 + sqrt{82335} iright)^{frac{2}{3}} – left(464 sqrt[3]{18} + left(1 – sqrt{3} iright) left(24 + sqrt[3]{12} left(1 – sqrt{3} iright) sqrt[3]{1503 + sqrt{82335} i}right) sqrt[3]{1503 + sqrt{82335} i}right)^{2}right)$$
=

-6.15360449013352 + 1.73343619316052e-28*i

$$y_{3} = -2 + frac{58}{3 left(- frac{1}{2} + frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}} + left(- frac{1}{2} + frac{sqrt{3} i}{2}right) sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}$$
=
$$frac{1}{left(1 – sqrt{3} iright) sqrt[3]{1503 + sqrt{82335} i}} left(- frac{116 sqrt[3]{18}}{3} + frac{1}{12} left(-24 + left(-1 + sqrt{3} iright) sqrt[3]{18036 + 12 sqrt{82335} i}right) left(1 – sqrt{3} iright) sqrt[3]{1503 + sqrt{82335} i}right)$$
=

-6.86684822099146 – 0.e-22*i

$$x_{4} = 41 – left(-2 + frac{58}{3 sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}} + sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}right)^{2}$$
=
$$41 – frac{1}{36 left(1503 + sqrt{82335} iright)^{frac{2}{3}}} left(116 sqrt[3]{18} – 12 sqrt[3]{1503 + sqrt{82335} i} + sqrt[3]{12} left(1503 + sqrt{82335} iright)^{frac{2}{3}}right)^{2}$$
=

-4.92173165585539 + 2.39490105785129e-27*i

$$y_{4} = -2 + frac{58}{3 sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}} + sqrt[3]{frac{167}{2} + frac{sqrt{82335} i}{18}}$$
=
$$-2 + frac{58 sqrt[3]{18}}{3 sqrt[3]{1503 + sqrt{82335} i}} + frac{1}{6} sqrt[3]{18036 + 12 sqrt{82335} i}$$
=

6.77655750775092 – 0.e-22*i

x1 = -6.153604490133523
y1 = -6.866848220991456

x2 = 5.00000000000000
y2 = 6.00000000000000

x3 = -4.921731655855394
y3 = 6.776557507750922