symbolically solving nonlinear equations

Question

I have two nonlinear function with variable $x,y,z$ and parameters $p_1,p_2,p_3$.

$$2(x−p_1)+2(xy−p_3)y=0 ,\\ 2(y−p_2)+2(xy−p_3)x=0$$

what is the value of $x,y,z$ in terms of $p_1,p_2,p_3$?

Answer 1

There is no magica wand for solving systems of linear equations. In your case, you can express $x$ as a function of $y$ from the first equation: $$x=x(y)=\frac{p_1+p_3y}{1+y^2}$$ Then replace $x$ with $x(y)$ in the second equation. You will get an equation with only one variable which you can try to solve. However, the equation for $y$ will include fifth degree polynomials in $y$ and will not be solvable with non numeric ways in general...