Q.Minimize : $z = 8x + 10y$ subject to $2x + y \ge 7,\ 2x + 3y \ge 15,\ y \ge 2,\ x \ge 0,\ y \ge 0$.

To minimum $z=8x+10y$ subject to $2x+y\ge7,\ 2x+3y\ge15,\ y\ge2,\ x\ge0,\ y\ge0$: draw each constraint's boundary line, shade the half-plane it demands, and darken the common (feasible) region. Its corner points, found by solving each pair of boundary lines that meet there, are evaluated in the objective function $z$ (the Corner-Point Theorem guarantees the optimum of a linear objective over a convex polygon occurs at one of its vertices):

| Corner point $(x,y)$ | $z$ |

|---|---| …

Corner-Point Method: (1) convert every constraint to an equation and draw its line; (2) shade the feasible region common to all constraints; (3) locate every vertex of this region by solving pairs of boundary equations simultaneously; (4) evaluate the objective function $z$ at each vertex; (5) the largest value is the maximum an …

Missing a vertex (especially where two constraints meet away from either axis) understates the true optimum; evaluating $z$ correctly at every listed vertex and double-checking the vertex itself s …