Q.Solve the following linear programming problem graphically: Minimise $Z = 200x + 500y$ subject to the constraints: $x + 2y \ge 10$, $3x + 4y \le 24$, $x \ge 0,\ y \ge 0$.

The Graphical Method for Linear Programming

When a linear programming problem has just two decision variables, $x$ and $y$, you can solve it by drawing a picture. This is the graphical method, and it is the technique the CBSE Class-12 course expects you to use.

The idea

Each constraint is a linear inequality such as $2x + 3y \le 100$. On the $xy$-plane its boundary is a straight line, and the inequality picks one side of that line (a half-plane). The points that satisfy all the constraints at once form a single region — the feasible region. Your job is to find the point inside this region that makes the objective function $Z = ax + by$ largest or smallest.

The step-by-step procedure

1. Draw each constraint line. Replace every inequality by an equation and plot the line, usually by finding where it meets the axes.
2. Shade the correct side. Test a simple point (often the origin $(0,0)$) in the inequality. If it holds, the origin's side is the wanted half-plane; if not, take the other side. Always include the non-negativity conditions $x \ge 0,\ y \ge 0$, which keep you in the first quadrant.
3. Identify the feasible region. It is the overlap of all the shaded half-planes — the region satisfying every constraint together.
4. Find the corner (vertex) points. These are the points where the boundary lines cross. Read them off the graph or solve the two relevant lines simultaneously.
5. Evaluate $Z$ at every corner and pick the largest value (for a maximum) or the smallest (for a minimum).

Bounded vs unbounded

If the feasible region is a closed polygon (bounded), both the maximum and minimum are guaranteed and are found among the corners. If the region stretches to infinity (unbounded), a maximum or minimum may fail to exist — you then check whether $Z$ can be pushed indefinitely large or small in the open direction before concluding.

The bottom line

Graph the constraints, find the feasible region, list its corner points, and compare $Z = ax + by$ at each. The best corner is your optimal solution — a clean, visual route to the answer for any two-variable LP problem.

The graphical method for solving linear programming problems is the entire method taught in the NCERT Class 12 Linear Programming chapter, and "linear programming graphical method examples class 12" is one of the most searched topics ahead of CBSE board exams. This corner-point approach is also occasionally tested in JEE Main and select state CET papers involving optimization.