I have three different equations of the form below, with two variables, x and y, and all other letters are constants. I want an algorithm that solves for x and y under any real values of the constants. Thanks for any help!
$a_1x^3 + b_1x^2y + c_1xy^2 + d_1y^3 + e_1x^2 + f_1x + g_1y^2 + h_1y + k_1 = 0$
$a_2x^3 + b_2x^2y + c_2xy^2 + d_2y^3 + e_2x^2 + f_2x + g_2y^2 + h_2y + k_2 = 0$
$a_3x^3 + b_3x^2y + c_3xy^2 + d_3y^3 + e_3x^2 + f_3x + g_3y^2 + h_3y + k_3 = 0$
I'm hesitant to substitute to reduce from cubics to quadratics, because I don't know how to avoid missing a root in the process.
I found a link to a book by Sturmfels, but I have only high school math, and I was not able to identify an answer here: https://math.berkeley.edu/~bernd/cbms.pdf
I found this related discussion, but its algebraic manipulation between polar and cartesian expressions is more than I could create to solve something like this: Solving a system of two cubic equations