Solving system of equations with $3$ variables:

Question

I am looking for some detail about how the following system of equations is solved, I checked using wolfram and got the answer but I am not sure the details of how to get there.

$$x+y=1$$ $$xw+y=\frac{1}{2}$$ $$xw^2+y = \frac{1}{3}$$

and the solution is $w = \frac{1}{3}, x = \frac{3}{4}, y = \frac{1}{4}$

Answer 1

From the first equation, you get that $y=1-x$. So, you now have only two equations:$$\left\{\begin{array}{l}xw-x=-\frac12\\xw^2-x=-\frac23,\end{array}\right.$$which is equivalent to$$\left\{\begin{array}{l}x(w-1)=-\frac12\\x(w-1)(w+1)=-\frac23.\end{array}\right.$$But now, dividing the second equality by the first one gives you $w+1=\frac43$. So…

Answer 2

Subtract the second equation from the first equation \begin{eqnarray*} x(1-w)=\frac{1}{2} \end{eqnarray*} Subtract the third equation from the second equation \begin{eqnarray*} xw(1-w)=\frac{1}{6} \end{eqnarray*} So $w=?$ ....