Linear Programming is a full 12-mark question in Section B of ECZ Grade 12 Mathematics Paper 2. Candidates must translate word problem constraints into linear inequalities, plot boundary lines on a Cartesian grid, shade unrequired regions to expose the feasible region $R$, and optimize objective functions.
CRITICAL EXAM RULE:
In ECZ Mathematics, you MUST shade the unwanted (unrequired) region. The wanted feasible region $R$ must remain completely unshaded in the middle!
1. Translating Word Problem Phrases into Inequalities
Common ECZ Constraint Phrasings:
- "At least $x$ units" $\implies x \ge \text{value}$
- "Not more than $y$ items" $\implies y \le \text{value}$
- "Total items cannot exceed 50" $\implies x + y \le 50$
- "Number of type X must be more than type Y" $\implies x > y$
- "Non-negativity constraints" $\implies x \ge 0, y \ge 0$
2. Plotting Lines and Shading Unrequired Regions
- Find Axis Intercepts: Convert inequality $a x + b y \le c$ to equation $a x + b y = c$. Set $x = 0$ to find $y$-intercept; set $y = 0$ to find $x$-intercept.
- Draw Boundary Line:
- Use a solid line for $\le$ or $\ge$.
- Use a broken (dashed) line for $<$ or $>$.
- Test Point (0,0): Substitute $(0,0)$ into the inequality.
- If statement is TRUE, $(0,0)$ is in the wanted region $\implies$ shade the side opposite $(0,0)$.
- If statement is FALSE, $(0,0)$ is in the unwanted region $\implies$ shade the side containing $(0,0)$.
Worked Exam Example (Paper 2):
A baker makes $x$ loaves of white bread and $y$ loaves of brown bread per day under these conditions:
1. He bakes at least 10 loaves of white bread: $x \ge 10$
2. Total loaves cannot exceed 40: $x + y \le 40$
3. Number of brown loaves is at least half the white loaves: $y \ge \frac{1}{2} x$
(a) Write down the three inequalities.
$$x \ge 10, \quad x + y \le 40, \quad 2y \ge x$$
(b) Plotting and Shading:
Line 1: $x = 10$ (vertical solid line). Shade region to the left ($x < 10$).
Line 2: $x + y = 40$ (intercepts $(0,40)$ and $(40,0)$). Shade region above line ($x + y > 40$).
Line 3: $2y = x$ (passes through $(0,0)$ and $(20,10)$). Shade region below line.
The unshaded triangular polygon in the center is labeled $R$.
(c) Maximizing Profit:
If white bread yields K5 profit and brown bread yields K8 profit, the profit function is $P = 5x + 8y$.
Check integer corner vertices of region $R$: $(10,5)$, $(10,30)$, and $(26.6,13.3) \to (26,14)$.
At $(10,30)$: $P = 5(10) + 8(30) = 50 + 240 = \text{K290}$ (Maximum Profit).
Teacher Tip:
Remind students to always use a sharp 2H pencil for boundary lines and light diagonal hatching for unwanted regions so the letter $R$ stands out clearly.
Graph Paper Checklist:
Ensure graph scales on both $x$ and $y$ axes match the exact scale specified in the exam paper (e.g., "2 cm to represent 5 units on both axes").