Graphical Method of Solution

Once an LPP has been formulated with exactly two decision variables, it can be solved by an exact geometric procedure, because every linear constraint corresponds to a straight line dividing the coordinate plane into two half-planes, and the set of points satisfying ALL the constraints at once is simply the region common to all of these half-planes (intersected with the first quadrant, from $x\ge0,\ y\ge0$).

Step 1 -- Plot each constraint's boundary line. For a constraint such as $px+qy\le H$, first replace the inequality by the equality $px+qy=H$ and plot this straight line, most easily by finding its two intercepts: setting $y=0$ gives the $x$-intercept $x=H/p$, and setting $x=0$ gives the $y$-intercept $y=H/q$. The actual constraint $px+qy\le H$ includes this boundary line itself (since the inequality is not strict), so the line is drawn as a solid line, not a dashed one.

Step 2 -- Decide which half-plane satisfies the inequality. Every straight line divides the plane into two half-planes, and exactly one of them (together with the line itself) satisfies a given $\le$ or $\ge$ inequality. The standard way to decide which is to substitute a convenient test point not on the line -- the origin $(0,0)$ is used whenever the line does not pass through it, since substituting $x=0,y=0$ is the simplest possible check. If the test point satisfies the inequality, the half-plane containing that test point is the one required; if not, the OTHER half-plane is required. (If the boundary line happens to pass through the origin, any other convenient point not on the line, such as $(1,0)$ or $(0,1)$, is used instead.)

Step 3 -- Shade the region common to every constraint. Repeating Steps 1--2 for every constraint (including $x\ge0$, the region to the right of the $y$-axis, and $y\ge0$, the region above the $x$-axis) and shading only the region that lies within EVERY required half-plane simultaneously gives the feasible region: the set of every point $(x,y)$ that satisfies all the constraints of the LPP at once, drawn as a single shaded polygon in the first quadrant (or, for an unbounded region, an open shaded area, Section 4). …