Linear Equations and Inequalities

Maths, SSC-Punjab Class 9. This chapter is split into 39 topics and has 963 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Exercise 5.1 Q2 and Review Exercise 5: Inequalities 24 questions
2. Checking the Solution of an Equation 30 questions
3. Example 7: Maximum and Minimum of f(x, y) = 2x + 3y 21 questions
4. Corner Points (Vertices) of the Feasible Region 26 questions
5. Linear Programming: Computed Problems 28 questions
6. Evaluating the Objective Function at Corner Points 24 questions
7. Linear Inequality in Two Variables: Forms and Solutions 24 questions
8. Representing Equation Solutions on the Number Line 24 questions
9. Operations on Inequalities and the Order of an Inequality 28 questions
10. Procedure for Graphing a Linear Inequality in Two Variables 25 questions
11. Inequality Symbols and Forms 24 questions
12. Solution of Two Linear Inequalities in Two Variables 27 questions
13. Objective Function and Optimal Solution 23 questions
14. Review Exercise 5: Multiple Choice Questions 24 questions
15. Unit 5 Introduction and Learning Outcomes 23 questions
16. Exercise 5.1 Q3: Shading Solution Regions 26 questions
17. Example 3: The Graph of x + 2y < 6 23 questions
18. Equations with Brackets and Variables on Both Sides 26 questions
19. Testing Points in a System of Inequalities 22 questions
20. Solving Inequalities with Fractions (Example 2) 25 questions
21. Interval Notation for Solutions of Inequalities 24 questions
22. Example 4: Vertical and Horizontal Boundary Lines 25 questions
23. Solving Linear Inequalities in One Variable 27 questions
24. Linear Equation in One Variable 24 questions
25. Exercise 5.1 and Review: Solving Equations 23 questions
26. Exercise 5.2: Linear Programming Problems 26 questions
27. Half Planes and Boundary Lines 25 questions
28. Solution Sets and Testing Values of an Inequality 24 questions
29. Closed and Open Half Planes (Notes on Example 3) 25 questions
30. Problem Constraints, Non-negative Constraints and Decision Variables 24 questions
31. Exercise 5.1 Q4: Systems and Intersection of Boundary Lines 25 questions
32. Graphing Inequalities on the Real Line 25 questions
33. Intercepts of Boundary Lines 26 questions
34. Steps to Solve a Linear Equation 25 questions
35. Review Exercise 5: Solution Regions and Optimization Problems 24 questions
36. Solving Equations with Fractions 24 questions
37. Solving Linear Equations with Integer Coefficients 25 questions
38. Feasible Region and Feasible Solution 23 questions
39. Example 6: Feasible Region and Corner Points 22 questions

Sample questions: Linear Equations and Inequalities

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. In Fig. 5.7(a) of Example 6, the shaded region shows

From your book: Figure 5.7 (page 93) Full page Open book

From your book: Example 6 (page 70) Full page Open book

• A. the graph of y ≥ 0 only
• B. the graph of x ≥ 0 only
• C. the final feasible region with all four constraints
• D. the intersection of the closed half planes of x − y ≤ 3 and x + 2y ≤ 6(correct answer)

Answer: D.The book says the intersection of the two closed half planes (both on the origin side) is partially displayed in Fig. 5.7(a); Figs. 5.7(b) and (c) add y ≥ 0 and x ≥ 0, and Fig. 5.7(d) is the feasible region.

Q2. Which of the following is a corner point of the feasible region of x + 5y ≤ 8, 2x + 5y ≤ 12, x ≥ 0, y ≥ 0?

• A. (6, 0)(correct answer)
• B. (1, 1)
• C. (7, 1)
• D. (8, 0)

Answer: A.(6, 0) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q3. Which of the following is a corner point of the feasible region of x + 2y ≤ 15, 3x + 3y ≤ 24, x ≥ 0, y ≥ 0?

• A. (8, 5)
• B. (1, 1)
• C. (1, 7)(correct answer)
• D. (5, 8)

Answer: C.(1, 7) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q4. Which of the following is a corner point of the feasible region of 3x + 5y ≤ 18, x + 5y ≤ 10, 3x + 3y ≥ 5, x ≥ 0, y ≥ 0?

• A. (4, 6/5)(correct answer)
• B. (3, 3)
• C. (2, 1)
• D. (8, 3)

Answer: A.(4, 6/5) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q5. Which of the following is a corner point of the feasible region of 5x + y ≤ 12, 5x + 3y ≤ 30, x + 2y ≥ 2, x ≥ 0, y ≥ 0?

• A. (3, 2)
• B. (2, 1)
• C. (12/5, 0)(correct answer)
• D. (3, 4)

Answer: C.(12/5, 0) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q6. Which of the following is a corner point of the feasible region of x + 3y ≤ 15, 3x + 2y ≤ 24, x ≥ 0, y ≥ 0?

• A. (9, 0)
• B. (2, 3)
• C. (9, 9)
• D. (0, 0)(correct answer)

Answer: D.(0, 0) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q7. Which of the following is a corner point of the feasible region of 4x + y ≤ 15, 4x + 4y ≤ 20, 3x + y ≥ 4, x ≥ 0, y ≥ 0?

• A. (2, 1)
• B. (0, 9)
• C. (4, 4)
• D. (0, 4)(correct answer)

Answer: D.(0, 4) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q8. Which of the following is a corner point of the feasible region of x + y ≤ 12, x + 2y ≤ 14, x ≥ 0, y ≥ 0?

• A. (7, 1)
• B. (8, 4)
• C. (5, 8)
• D. (12, 0)(correct answer)

Answer: D.(12, 0) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q9. Which of the following is a corner point of the feasible region of x + y ≤ 8, x + 4y ≤ 24, 3x + 2y ≥ 5, x ≥ 0, y ≥ 0?

• A. (5, 6)
• B. (0, 6)(correct answer)
• C. (2, 3)
• D. (5, 8)

Answer: B.(0, 6) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.

Q10. Which of the following is a corner point of the feasible region of 3x + 4y ≤ 24, 3x + y ≤ 18, x + 2y ≥ 2, x ≥ 0, y ≥ 0?

• A. (2, 4)
• B. (6, 3)
• C. (6, 0)(correct answer)
• D. (4, 6)

Answer: C.(6, 0) is an intersection of two boundary lines and satisfies all constraints; the interior point lies on no boundary and the others violate a constraint.