Where do the points (-1,3) and (2,1) come from?

Question

I now understand how the green area is obtained from the set definition below in point 2, but how is that set itself constructed? More precisely, where do the points $(-1,3)$ and $(2,1)$ come from?

Answer 1

If $(x_1,x_2)$ is a point in $R^2$ that's better than $(2,2)$ wrt both objectives, it must be the case that $$\begin{align} 3x_1+x_2 &> 3(2) + (2)\\ -x_1 + 2x_2&>-(2)+ 2(2) \end{align}\tag a $$ Rewrite system (a) into the form: $$\begin{align} 3(x_1-2)+(x_2-2) &> 0\\ -(x_1-2) + 2(x_2- 2)&>0 \end{align}\tag b $$ If we write $\beta$ for the LHS of the first inequality in (b) and $\alpha$ for the LHS of the second inequality, then we are requiring $\alpha>0$ and $\beta>0$. Solve the LHS of system (b) for $x_1$ and $x_2$ in terms of $\alpha$ and $\beta$ to obtain a characterization of all points $(x_1,x_2)$ better than $(2,2)$: $$ \begin{align} x_1-2=-\frac17\alpha +\frac27\beta\\ x_2-2=\frac37\alpha+\frac17\beta \end{align}\tag c $$ After relabeling $\lambda_1:=\frac\alpha7$ and $\lambda_2:=\frac\beta7$, the equations (c) are exactly the ones in point 2.