In ECZ Mathematics Paper 2 Section B, Linear Programming carries 8 full marks. The single most fatal mistake candidates make is shading the wanted region. The instructions explicitly command: "By shading the UNWANTED region, show the region R that satisfies all inequalities." Leaving $R$ unshaded is mandatory!
Train learners to translate English inequality keywords into mathematical symbols methodically: "at most" $\le$, "not more than" $\le$, "at least" $\ge$, "not less than" $\ge$. Always remind candidates to include non-negativity constraints $x \ge 0, y \ge 0$ when items cannot be negative in real life.
Table of Contents
- 1. The Translation Dictionary: English to Mathematical Inequalities
- 2. Plotting Boundary Lines (Solid vs Dashed)
- 3. Testing Points & Shading the UNWANTED Region
- 4. The Objective Function: Maximizing Profit or Minimizing Cost
- 5. Step-by-Step Worked 8-Mark Past Examination Question
- 6. Finding Optimal Integer Solutions within Region R
- 7. Frequently Asked Questions
1. The Translation Dictionary: English to Mathematical Inequalities
Linear programming questions present real-world scenarios involving transport, farming, or factory manufacturing. Use this master translation table:
| English Word Expression | Mathematical Meaning | Symbol | Concrete Exam Example |
|---|---|---|---|
| "At least" / "No fewer than" | Minimum value; greater than or equal to | $\ge$ | At least 5 bags of fertilizer: $x \ge 5$ |
| "At most" / "Not more than" | Maximum ceiling; less than or equal to | $\le$ | Not more than 40 passengers: $y \le 40$ |
| "Exceeds" / "Strictly greater than" | More than | $>$ | Number of boys exceeds girls: $x > y$ |
| "Does not exceed" | Less than or equal to | $\le$ | Total expenditure does not exceed K5,000: $ax + by \le 5000$ |
| "At least twice as many $y$ as $x$" | $y$ is greater than or equal to $2x$ | $y \ge 2x$ | Bags of wheat $y$ at least double maize $x$: $y \ge 2x$ |
2. Plotting Boundary Lines (Solid vs Dashed)
To graph any inequality on a Cartesian plane, first replace the inequality symbol with an equals sign ($=$) to form the boundary line equation:
Steps to Plot Boundary Lines:
- Find the intercepts: Set $x = 0$ to find the $y$-intercept; set $y = 0$ to find the $x$-intercept.
- Plot the two points and connect them with a straight ruler line.
- Line Style Rule:
- For $\le$ or $\ge$, draw a solid line (points on the line are included in the solution).
- For $<$ or $>$, draw a broken (dashed) line (points on the line are excluded).
- Label each line clearly with its equation (e.g., $2x + 3y = 60$).
3. Testing Points & Shading the UNWANTED Region
This is the core convention of the Examination Council of Zambia:
The Test Point Strategy (Use $(0,0)$):
Pick a simple test point not lying on the line—the origin $(0,0)$ is ideal:
- Substitute $x = 0, y = 0$ into the original inequality.
- If $(0,0)$ makes the inequality TRUE, then $(0,0)$ is in the WANTED region. Therefore, you must SHADE THE OPPOSITE SIDE (the side without $(0,0)$).
- If $(0,0)$ makes the inequality FALSE, then $(0,0)$ is in the UNWANTED region. Therefore, you must SHADE THE SIDE CONTAINING $(0,0)$.
- When all inequalities are shaded, the clean, unshaded white area is labeled with a bold capital R (Feasible Region).
4. The Objective Function: Maximizing Profit
Once region $R$ is defined, the final sub-question asks for optimal allocation (e.g., maximum profit):
By the Fundamental Theorem of Linear Programming, the optimal maximum or minimum value always occurs at one of the corner vertices (intersections) of the feasible polygon $R$.
5. Step-by-Step Worked 8-Mark Past Examination Question
A transporter has two types of trucks: Small trucks ($x$) and Large trucks ($y$).
- A small truck carries 20 tonnes of maize; a large truck carries 50 tonnes. The transporter must deliver at least 400 tonnes of maize.
- The total number of trucks available cannot exceed 16.
- The number of small trucks must be at least 4.
- The number of large trucks must not be more than 10.
(a) Write down four inequalities representing the conditions above. [3 Marks]
(b) Using a scale of $2\text{ cm}$ to represent 2 units on both axes, shade the unwanted regions to show region $R$. [3 Marks]
(c) If the hire cost of a small truck is K2,000 and a large truck is K4,000, find the number of trucks of each type that minimizes total cost. [2 Marks]
Full Step-by-Step Solution:
Part (a): Formulating Inequalities
2. Total trucks constraint: $\mathbf{x + y \le 16}$
3. Minimum small trucks: $\mathbf{x \ge 4}$
4. Maximum large trucks: $\mathbf{y \le 10}$
(Non-negativity: $x \ge 0, y \ge 0$ implicitly satisfied)
[3 Marks Awarded]
Part (b): Boundary Lines for Graphing
• Line 2 ($x + y = 16$): Intercepts $(0,16)$ and $(16,0)$. Test $(0,0)$: $0 \le 16$ (TRUE). Shade side above the line away from $(0,0)$.
• Line 3 ($x = 4$): Vertical line through $x=4$. Shade left of the line ($x < 4$).
• Line 4 ($y = 10$): Horizontal line through $y=10$. Shade above the line ($y > 10$).
The unshaded polygon is labeled R. [3 Marks Awarded]
Part (c): Minimizing Cost Function: $C = 2000x + 4000y$
• Vertex $(4, 10) \implies C = 2000(4) + 4000(10) = 8000 + 40000 = \text{K}48,000$
• Point $(5, 6) \implies 2(5) + 5(6) = 10 + 30 = 40$: $C = 2000(5) + 4000(6) = 10000 + 24000 = \text{K}34,000$
• Point $(10, 4) \implies 2(10) + 5(4) = 40$: $C = 2000(10) + 4000(4) = 20000 + 16000 = \text{K}36,000$
• Point $(15, 2) \implies 2(15) + 5(2) = 40$: $C = 2000(15) + 4000(2) = 30000 + 8000 = \text{K}38,000$
The minimum cost is achieved at 5 Small trucks and 6 Large trucks (Total: K34,000).
Final Answer: 5 Small Trucks, 6 Large Trucks [2 Marks]
6. Finding Optimal Integer Solutions within Region R
In real-world linear programming problems, you cannot buy 2.5 trucks or harvest 7.3 bags of maize. If a geometric corner vertex results in fractional decimals (e.g., $(6.4, 3.2)$):
- Inspect the integer grid coordinates $(x, y)$ strictly inside or on the solid boundaries of region $R$ closest to that vertex.
- Test those candidate integer points in your objective profit/cost equation.
- Select the integer pair that fulfills all constraints and maximizes/minimizes the target value.
7. Frequently Asked Questions
What happens if I shade the wanted region instead of the unwanted region?
ECZ marking schemes award zero marks for the region $R$ if the candidate shades the inside rather than the unwanted outside. Always follow the explicit exam instruction: "shade the unwanted region".
Can a line pass through the origin $(0,0)$? How do I test it?
If an inequality has no constant term (such as $y \ge 2x$), the line passes directly through $(0,0)$. In this case, test a point off the line, such as $(1,0)$ or $(0,2)$, to determine which half-plane to shade.
How do I draw a search line for optimization?
Set the profit equation to a convenient constant (e.g., $2000x + 4000y = 16000 \implies x + 2y = 8$). Draw this dashed line, then slide your ruler parallel across region $R$ to find the last vertex it touches before exiting the region for maximum profit.