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Linear programming — KCSE Mathematics

KCSE Mathematics · 107 practice questions · 3 syllabus objectives · 3 revision lessons

33 easy38 medium36 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Form linear inequalities based on real life situations

Represent linear inequalities on a graph and solve them

Solve and interpret the optimum solution using the objective function

Revision Notes

Concise lesson notes for Linear programming, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Linear Inequalities from Real-Life Situations

Linear inequalities are mathematical expressions that show the relationship between variables with an inequality sign. To form linear inequalities from real-life situations, follow these steps:

  1. Identify the variables involved in the situation.
  2. Determine the constraints or limits based on the scenario.
  3. Express the relationship using inequality symbols (>, <, ≥, ≤).

Example 1: A farmer has 100 meters of fencing to create a rectangular pen for sheep. Let x be the length and y be the width of the pen.

  • The perimeter constraint can be expressed as: 2x + 2y ≤ 100.

Example 2: A school can accommodate a maximum of 300 students in a hall. Let s represent the number of students.

  • The inequality can be written as: s ≤ 300.

These inequalities can be graphed to visualize feasible solutions based on the constraints defined.

Key points to remember

  • Identify variables in real-life scenarios.
  • Determine constraints based on the situation.
  • Use inequality symbols to express relationships.

Worked example

A store sells pencils at 20 Ksh each and erasers at 30 Ksh each. If the budget is 600 Ksh, form an inequality: 20x + 30y ≤ 600.

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Lesson 2: Graphing Linear Inequalities

Objective: Represent linear inequalities on a graph and solve them

To represent linear inequalities graphically, follow these steps:

  1. Convert the inequality to an equation: Replace the inequality sign with an equals sign to find the boundary line.
  2. Graph the boundary line: Use a solid line for ≤ or ≥ and a dashed line for < or >.
  3. Choose a test point: Select a point not on the line (commonly (0,0)) to determine which side of the line to shade.
  4. Shade the appropriate area: If the test point satisfies the inequality, shade the side containing that point; otherwise, shade the opposite side.

For example, consider the inequality: 2x + 3y < 6.

  • Convert to equation: 2x + 3y = 6.
  • Graph the line with a dashed line.
  • Test point (0,0): 2(0) + 3(0) < 6 is true, so shade the region containing (0,0).

This graphical representation helps visualize the solution set of the inequality, which is crucial in linear programming.

  • Convert inequalities to equations for boundary lines.
  • Use solid lines for ≤ or ≥, dashed for < or >.
  • Test points help determine the shaded region.
  • Shaded area represents all solutions of the inequality.
  • Graphing aids in visualizing constraints in linear programming.

Graph the inequality x + y ≥ 4.

  • Convert to equation: x + y = 4.
  • Graph with a solid line.
  • Test point (0,0): 0 + 0 ≥ 4 is false, shade the opposite side.
Lesson 3: Solving Linear Programming Problems

Objective: Solve and interpret the optimum solution using the objective function

In linear programming, we aim to maximize or minimize an objective function subject to constraints. The optimum solution occurs at the vertices of the feasible region. To solve a linear programming problem, follow these steps:

  1. Formulate the objective function: Identify what you want to maximize or minimize, such as profit or cost.
  2. Identify constraints: These are the inequalities that restrict the solution.
  3. Graph the constraints: Plot the inequalities on a graph to find the feasible region.
  4. Locate vertices: The optimum solution will be at a vertex of the feasible region.
  5. Evaluate the objective function: Calculate the value of the objective function at each vertex to determine the optimum solution.

For example, consider the objective function: Maximize Z = 3x + 2y subject to the constraints:

  • x + y ≤ 4
  • x ≥ 0
  • y ≥ 0

After graphing the constraints, the feasible region is identified, and its vertices are (0, 0), (0, 4), and (4, 0). Evaluating Z:

  • At (0, 0): Z = 0
  • At (0, 4): Z = 8
  • At (4, 0): Z = 12

The optimum solution is at (4, 0) with a maximum value of Z = 12.

  • Identify the objective function to maximize or minimize.
  • Graph constraints to find the feasible region.
  • Evaluate the objective function at each vertex.
  • Optimum solutions occur at feasible region vertices.
  • Interpret the optimum solution in context.

Maximize Z = 5x + 4y subject to x + 2y ≤ 8, x ≥ 0, y ≥ 0. Optimum solution is at (0, 4) with Z = 16.

Sample Questions

Read 3 questions and answers free. Sign up to access all 107 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER2 marks

A company produces two types of drinks: soda (x) and juice (y). Each soda can is sold for Ksh 50 and each juice carton for Ksh 70. The company aims to maximize its revenue. (a) Write down the objective function for the total revenue. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
R = 50x + 70y (2 mks)
2
easySHORT ANSWER2 marks

A factory produces two products: chairs (x) and tables (y). Each chair requires 2 hours of assembly time and each table requires 3 hours. The factory has a maximum of 24 hours available for assembly each week. (a) State the inequality that represents the assembly time constraint. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
2x + 3y ≤ 24 (2 mks)
3
easySHORT ANSWER2 marks

A café sells two types of drinks: coffee (x) and tea (y). Each coffee requires 200 ml of water and each tea requires 150 ml. The total water available is 15 litres. (a) Write down the inequality representing the water constraint. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
0.2x + 0.15y ≤ 15 (2 mks)
4

A farmer grows two types of crops: maize (x) and beans (y). Each maize plant requires 2 m² of land, while each bean plant requires 1.5 m². The total land available is 100 m². (a) State the inequality that represents the land constraint. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Linear programming" cover?

Formation of linear inequalities from real life; graphical solutions; optimisation using objective function

How many practice questions are available for Linear programming?

HighMarks has 107 Linear programming practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Linear programming for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

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