Q.Solve graphically: Minimize $Z = 3x + 2y$ subject to $x + y \ge 10,\ 2x + y \ge 15,\ x + 2y \ge 12,\ x, y \ge 0$.

Constraints: $x+y\ge10,\ 2x+y\ge15,\ x+2y\ge12,\ x,y\ge0$.

At $y=0$: the three lines need $x\ge10,\ x\ge7.5,\ x\ge12$; binding is $x\ge12$, vertex $(12,0)$.

At $x=0$: the three lines need $y\ge10,\ y\ge15,\ y\ge6$; binding is $y\ge15$, vertex $(0,15)$.

$(x+y=10)\cap(2x+y=15)$: subtracting, $x=5,\ y=5$, giving $(5,5)$; check $x+2y\ge12$: $5+10=15\ge12$ ✓ feasible.

$(x+y=10)\cap(x+2y=12)$: subtracting, $y=2,\ x=8$, giving $(8,2)$; check $2x+y\ge15$: $16+2=18\ge15$ ✓ feasible. …

For three ≥ constraints, find all three pairwise intersections, test each against the remaining constraint, …

Keeping the infeasible intersection $(6,3)$ as a vertex without checking it against $x+y\ge10$, …