We are given two quadratic equations $$ 0= a x^2 + b x + c + d y $$ and $$ 0= A y^2 + B y + C + D x $$ with real constants $a,b,c,d$ and $A, B, C, D$.
Under which conditions on the constants do (one or several) real solutions $(x,y)$ exist?
For one variable ($d=D=0$) we obtain two uncoupled equations that are solved by the standard solution formula $$x= \frac{-b\pm\sqrt{b^2-4 a c}}{2a}$$ (or the equivalent with capital constants) such that real solutions exist if and only if the discriminant is non-negative, $$b^2-4 a c\geq 0$$. Is there any such criterion for two coupled equations? I understand that one can solve one equation, say the second for $y$ and substitute into the first but I am looking for a more general understanding to I can generalize to larger numbers of
linearly coupled quadratic equations, and answer under which conditions solutions exist.