How many solutions this a multidimensional system have? How many solutions have this system of equations:
\begin{array}{l}
\ 3x^2+y+2xy^2-3 = 0 \\
\ x + 2yx = 0
\end{array}
The more general case is when we want to solve two nonlinear algebraic equations for two unknowns
\begin{array}{l}
\ {A_1}{x^2} + {B_1}xy + {C_1}{y^2} + {D_1}x + {E_1}y + {F_1} = 0 \\
\ {A_2}{x^2} + {B_2}xy + {C_2}{y^2} + {D_2}x + {E_2}y + {F_2} = 0
\end{array}

There are two roots of the second equation.
$x=0$ substituted into the first equation gives $ y=3.$
$y=-\frac12$ substituted into the first equation gives two solutions from the quadratic resulting in:
$$ x = \dfrac{-3 \pm \sqrt{82}}{2}  $$
So, in all 3 different roots and 3 solutions.
