I would like to know how to algebraically (without graphing) find coordinates delimiting the solution region of a system of linear inequalities. For example:
$$ \left\{ \begin{array}{c} x\ge2 \\ y\ge3 \\ y\le-x+5 \\ x\le3 \end{array} \right. $$
There are 5 intersections in total: $(2, 3)$, $(2, 1)$, $(3, 2)$, $(3, 1)$ and $(4, 1)$, but the last is not part of the solution region. I know I can find all the intersections, but how can I determine algebraically (without graphing) which ones of them are making the solution region?
Thanks.
I would say that the easiest is to first establish bounds on $x$ and $y$ separately. Here, you have $2\leqslant x\leqslant 3$ and $-x+5\geqslant y\geqslant 3\implies x\leqslant 2$. But then, we already have $x=2$. For the bound on $y$, notice that $3\leqslant y\leqslant -x+5=3\implies y=3$. Therefore, the only point that can satisfy the system of inequalities is $P=(2,3)$; it is easy to check that $P$, indeed, fulfills every inequality.