Solve {l}{x^3+5y=0}{x-3y^2=0}

$$\left. \begin{array} { l } { x ^ { 3 } + 5 y = 0 } \\ { x - 3 y ^ { 2 } = 0 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=0\text{, }y=0$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}}{3}\approx 1.528142136\text{, }y=-\frac{3^{\frac{2}{5}}\sqrt[5]{5}}{3}\approx -0.713709123$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}e^{\frac{2\pi i}{5}}}{3}\approx 0.47222189+1.453349536i\text{, }y=\frac{3^{\frac{2}{5}}\sqrt[5]{5}e^{\frac{\pi i}{5}}}{3}\approx 0.577402809+0.419507697i$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}e^{\frac{4\pi i}{5}}}{3}\approx -1.236292958+0.898219411i\text{, }y=\frac{3^{\frac{2}{5}}\sqrt[5]{5}e^{\frac{7\pi i}{5}}}{3}\approx -0.220548248-0.678777712i$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}e^{\frac{6\pi i}{5}}}{3}\approx -1.236292958-0.898219411i\text{, }y=\frac{3^{\frac{2}{5}}\sqrt[5]{5}e^{\frac{3\pi i}{5}}}{3}\approx -0.220548248+0.678777712i$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}e^{\frac{8\pi i}{5}}}{3}\approx 0.47222189-1.453349536i\text{, }y=\frac{3^{\frac{2}{5}}\sqrt[5]{5}e^{\frac{9\pi i}{5}}}{3}\approx 0.577402809-0.419507697i$$

Solve for x, y

$$x=0\text{, }y=0$$

$$x=\frac{3^{\frac{4}{5}}\times 5^{\frac{2}{5}}}{3}\approx 1.528142136\text{, }y=-\frac{3^{\frac{2}{5}}\sqrt[5]{5}}{3}\approx -0.713709123$$