# Solve system equation $\left\{ \begin{array}{l} xy - x - y = 1\\ 4{x^3} - 12{x^2} + 9x = - {y^3} + 6y + 7 \end{array} \right.$

Solve system equation : $$\left\{ \begin{array}{l} xy - x - y = 1\\ 4{x^3} - 12{x^2} + 9x = - {y^3} + 6y + 7 \end{array} \right. ,\quad (x,y\in\mathbb{R}).$$ My solution begin with : Set $z=x-1$ we have : $$\left\{ \begin{array}{l} yz=z+2\\ 4z^3-3z+y^3-6y-6=0 \end{array} \right.\Leftrightarrow \left\{ \begin{array}{l} yz=z+2\\ 4z^3-3y^2z+y^3=0 \end{array} \right.$$ I want to have a difference solution.

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What do you mean by "a difference solution"? – Gerry Myerson Sep 4 '13 at 12:10

Using the straightforward approach, I find $x=\frac {y+1}{y-1}$ from the first equation and substitute it in the second one $$\frac {{y}^{6}-3\,{y}^{5}-3\,{y}^{4}+11\,{y}^{3}-6\,{y}^{2}+32}{ \left( y-1 \right) ^{3}}=0.$$ The keypoint is the next step $$y^6-3y^5-3y^4+11y^3-6y^2+32 =(y^2-y+2)(y^2-y-4)^2.$$ The rest is quite simple: $$\left \{ y=\frac 1 2 \pm \frac {\sqrt{7}i } 2, x= \frac {\frac 1 2 \pm \frac {\sqrt{7}i } 2 +1} {\frac 1 2 \pm \frac {\sqrt{7}i } 2 -1}\right \}$$ or $$\left\{ x=5/4-1/4\,\sqrt {17},y=1/2-1/2\,\sqrt {17} \right\}$$ or $$\left\{ x=5/4+1/4\,\sqrt {17},y=1/2+1/2\,\sqrt {17} \right\} .$$
The Maple command $$SolveTools:-PolynomialSystem([x*y-x-y = 1, 4*x^3-12*x^2+9*x = -y^3+6*y+7], [x, y]);$$ produces the same result. – user64494 Sep 4 '13 at 11:40