Solve {l}{x+y+z=5}{x^{2}+y^{2}+z^2=21}{x^3+y^3+z^3=80}

$$\left. \begin{array} { l } { x + y + z = 5 } \\ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 21 } \\ { x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 80 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{\sqrt{57}\sqrt{-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+1}-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 1.519969372\text{, }y=\frac{-\sqrt{57}\sqrt{-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+1}-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx -0.773387412\text{, }z=\frac{2\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 4.25341804$$

$$x=\frac{\sqrt{57}\sqrt{-\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 4.25341804\text{, }y=\frac{-\sqrt{57}\sqrt{-\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 1.519969372\text{, }z=\frac{-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx -0.773387412$$

$$x=\frac{\sqrt{57}\sqrt{\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 4.25341804\text{, }y=\frac{-\sqrt{57}\sqrt{\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx -0.773387412\text{, }z=\frac{\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 1.519969372$$

$$x=\frac{-\sqrt{57}\sqrt{-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+1}-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx -0.773387412\text{, }y=\frac{\sqrt{57}\sqrt{-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+1}-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 1.519969372\text{, }z=\frac{2\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 4.25341804$$

$$x=\frac{-\sqrt{57}\sqrt{-\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 1.519969372\text{, }y=\frac{\sqrt{57}\sqrt{-\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 4.25341804\text{, }z=\frac{-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx -0.773387412$$

$$x=\frac{-\sqrt{57}\sqrt{\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx -0.773387412\text{, }y=\frac{\sqrt{57}\sqrt{\sqrt{3}\sin(\frac{2\arccos(\frac{25\sqrt{19}}{722})}{3})-3\left(\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}-\left(\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})\right)^{2}+4}+\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+10}{6}\approx 4.25341804\text{, }z=\frac{\sqrt{57}\sin(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})-\sqrt{19}\cos(\frac{\arccos(\frac{25\sqrt{19}}{722})}{3})+5}{3}\approx 1.519969372$$