Solve two non linear polynomial equations with two variables (specific problem)

Question

I have the following two equations:

\begin{eqnarray} y(y-x-p) = x(y+3x-3p) \\ (p-x-y)^2 = k x y^3 \end{eqnarray}

where $p,k$ are constants known. How do I solve this for $x,y$? WolframAlpha provides

this solution.

I want to do it "by hand".

Answer 1

Hint : Let $P=x+y-p$, now note that the first equation can be rewritten as \begin{eqnarray*} y(P-2x)=x(3P-2y). \end{eqnarray*} Should be a doddle from here ?

Answer 2

Hint:   the first equation can be written as a quadratic in $y\,$, then solving it gives the factorization:

$$ 0 = y(y-x-p) - x(y+3x-3p) = y^2 - 2 x y - 3 x^2 + 3 p x - p y = (y - 3 x) (y + x -p) $$