I have $3$ equations and $2$ unknowns $x,y$ as following: $$ \left\{ \begin{array}{c} (\Delta_{11}*y^2 + \Delta_{12}*y + \Delta_{13})x^2 + (\Delta_{21}*y^2 + \Delta_{22}*y + \Delta_{23})x + \Delta_{31}*y^2 + \Delta_{32}*y + \Delta_{33} = 0 \\ (\Omega_{11}*y^2 + \Omega_{12}*y + \Omega_{13})x^2 + (\Omega_{21}*y^2 + \Omega_{22}*y + \Omega_{23})x + \Omega_{31}*y^2 + \Omega_{32}*y + \Omega_{33} = 0 \\ (\Gamma_{11}*y^2 + \Gamma_{12}*y + \Gamma_{13})x^2 + (\Gamma_{21}*y^2 + \Gamma_{22}*y + \Gamma_{23})x + \Gamma_{31}*y^2 + \Gamma_{32}*y + \Gamma_{33} = 0 \\ \end{array} \right. $$ where $\Delta_{i,j}$, $\Omega_{i,j}$ and $\Gamma_{i,j}$ are known values. Is it possible to solve this system of equations for $x,y$?
Thank you very much for your help!
HINT.-When you have a linear system in $x^2, x$ such as $$\begin{cases}A_1x^2+A_2x+A_3=0\\B_1x^2+B_2x+B_3=0\\C_1x^2+C_2x+C_3=0\end{cases}$$ you can see it as a system in three unkowns $x^2,x,w$ so you have $$\begin{pmatrix}A_1&A_2&A_3\\B_1&B_2&B_3\\C_1&C_2&C_3\end{pmatrix}\cdot\begin{pmatrix}x^2\\x\\w\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}$$
Now, since you have the obvious solution $(0,0,0)$ besides of the main solutions the solution of this system is not unique hence the determinant of the matrix of coefficients must be equal to $0$.
This gives a polynomial equation in $y$ of degree $6$.After solving this equation you have for each root a quadratic equation giving the corresponding values of $x$.