Q.In order to supplement daily diet, a person wishes to take some X and some Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligrams per tablet) are given as below: Tablet X contains 6 mg iron, 3 mg calcium, 2 mg vitamin; Tablet Y contains 2 mg iron, 3 mg calcium, 4 mg vitamin. The person needs at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Re 1 respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?

Graphical method. Let $x,y$ be the number of X and Y tablets. Minimise cost $Z=2x+y$.

Constraints (need at least the stated amounts):

with $x,y\ge 0$.

Feasible corner points: $(0,9),\ (1,6),\ (6,1),\ (8,0)$.

$Z(0,9)=9,\quad Z(1,6)=8,\quad Z(6,1)=13,\quad Z(8,0)=16.$ …

Set up

Let $x$ = number of X tablets and $y$ = number of Y tablets, $x,y\ge 0$. X costs Rs $2$, Y costs Re $1$, so minimise

The person needs at least the stated amounts, so the nutrient constraints are "$\ge$":

Corner points

The feasible region is unbounded (above-right of the three lines). Its vertices:

• $3x+y=9$ meets the $y$-axis at $(0,9)$ — check $x+y=9\ge 7$ (ok), $x+2y=18\ge 8$ (ok).
• $3x+y=9 \cap x+y=7$: subtracting gives $2x=2,\ x=1,\ y=6$ → $(1,6)$.
• $x+y=7 \cap x+2y=8$: subtracting gives $y=1,\ x=6$ → $(6,1)$.
• $x+2y=8$ meets the $x$-axis at $(8,0)$ — the other constraints hold (ok).

The crossing $3x+y=9 \cap x+2y=8$ gives $(2,3)$, but $x+y=5<7$, so it lies outside the region — discard it.

Evaluate the cost …

Method: Graphical (Corner-Point) Method for a Two-Variable LPP

Once a problem is written as "optimise $Z = ax + by$ subject to linear inequalities in $x$ and $y$," this is the standard CBSE technique for finding the optimum.

Steps

Step 1: Draw every constraint as a line.

Replace each inequality by an equation and plot the line from its axis intercepts. Include $x = 0$ and $y = 0$.

Step 2: Shade the correct half-plane.

Test the origin $(0,0)$ in each inequality: if it is satisfied, keep the origin's side; if not, take the other side. Non-negativity confines you to the first quadrant.

Step 3: Identify the feasible region.

It is the single region where all the shaded half-planes overlap.

Step 4: Find the corner points.

Each vertex is the intersection of two boundary lines, found by solving those two equations together. Keep a candidate only if it satisfies every other constraint -- an intersection that breaks a third inequality is not a corner of the region.

Step 5: Evaluate $Z$ at each corner. …

Common Mistakes

Mistake 1: Using the infeasible crossing $(2,3)$ as a corner.

Why it's wrong: $(2,3)$ lies on two of the lines but fails the calcium constraint ($x+y=5<7$), so it is outside the feasible region. Correct approach: reject it and use the genuine vertices $(0,9),(1,6),(6,1),(8,0)$.

Mistake 2: Assuming the smallest corner value is automatically the minimum on an unbounded region. …