Question

Using graphical method
Find the corner points for
$2x+5y\le 25$
$6x+5y\le 45$
$x\ge 0,y\ge 0$

Answer 1

Some corner points can usually found by inspection. In this case, we can see $A=(0,0)$ and $D(0,5)$. Some corner points may require some work with boundary lines (uses equations of boundaries not the inequalities giving the regions).
Point C
System: $2x+5y=\mathrm{25...}\left(1\right)$
$6x+5y=\mathrm{45.....}\left(2\right)$
$\left(1\right)-\left(2\right)$ $R-4x=-20$
$\Rightarrow$ $x=5$
If $x=5$, then from (1) or (2)
$y=3$
Point B:
System: $t=\mathrm{0.....}\left(1\right)$
$6x+5y=\mathrm{45....}\left(2\right)$
Solve by substitution:
$\Rightarrow$ $6x+5\left(0\right)=45=7.5$
The corner points for example $7$ are $(0,0),(0,5),(7.5,0)$ and $5,3)$
Convex sets and corner points lead us to a method for solving certain linear programming problems.