Chapter 5: Linear Equations and Inequalities

Mathematics

Class 92025–26 SNC6 Questions SolvedEnglish Medium

Exercise 5.2 — Class 9 Mathematics Solutions

All 6 questions below are worked step by step — every step clearly worked out with mathematical reasons — for the Punjab Board 9th class Mathematics textbook.

Exercise 5.2

6 Questions

Quick Solution Step by Step

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Question 1

Medium

Maximize $f (x, y) = 2 x + 5 y$ ; subject to the constraints: $2 y - x \leq 8, x - y \leq 4, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} 2 y - x & \leq 8, x - y \leq 4, x, y \geq 0 \\ Adding: y & = 12 \\ x & = 16 \\ Corner points: & O (0, 0), C (4, 0), E (16, 12), B (0, 4) \\ f (0, 0) & = 0 \\ f (4, 0) & = 8 \\ f (0, 4) & = 20 \\ f (16, 12) & = 32 + 60 \\ = 92 \\ Maximum: f (16, 12) & = 92 \end{matrix}

Step by Step

Question 2

Medium

Maximize $f (x, y) = x + 3 y$ ; subject to the constraints: $2 x + 5 y \leq 30, 5 x + 4 y \leq 20, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} 2 x + 5 y & \leq 30, 5 x + 4 y \leq 20, x, y \geq 0 \\ Corner points: & O (0, 0), C (4, 0), D (0, 5) \\ f (0, 0) & = 0 \\ f (4, 0) & = 4 \\ f (0, 5) & = 15 \\ Maximum: f (0, 5) & = 15 \end{matrix}

Step by Step

Question 3

Medium

Maximize $z = 2 x + 3 y$ ; subject to the constraints: $2 x + y \leq 4, 4 x - y \leq 2, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} 2 x + y & \leq 4, 4 x - y \leq 2, x, y \geq 0 \\ Adding: 6 x & = 6 \\ x & = 1 \\ y & = 2 \\ Corner points: & O (0, 0), C (\frac{1}{2}, 0), E (1, 2), B (0, 4) \\ z (0, 0) & = 0 \\ z (\frac{1}{2}, 0) & = 1 \\ z (1, 2) & = 8 \\ z (0, 4) & = 12 \\ Maximum: z (0, 4) & = 12 \end{matrix}

Step by Step

Question 4

Medium

Minimize $z = 2 x + y$ ; subject to the constraints: $x + y \geq 3, 7 x + 5 y \leq 35, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} x + y & \geq 3, 7 x + 5 y \leq 35, x, y \geq 0 \\ Corner points: & A (3, 0), C (5, 0), D (0, 7), B (0, 3) \\ z (3, 0) & = 6 \\ z (5, 0) & = 10 \\ z (0, 7) & = 7 \\ z (0, 3) & = 3 \\ Minimum: z (0, 3) & = 3 \end{matrix}

Step by Step

Question 5

Medium

Maximize the function defined as; $f (x, y) = 2 x + 3 y$ subject to the constraints: $2 x + y \leq 10, x + 2 y \leq 14, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} 2 x + y & \leq 10, x + 2 y \leq 14, x, y \geq 0 \\ (4 x + 2 y) - (x + 2 y) & = 20 - 14 \\ x & = 2 \\ y & = 6 \\ Corner points: & O (0, 0), A (5, 0), E (2, 6), D (0, 7) \\ f (0, 0) & = 0 \\ f (5, 0) & = 10 \\ f (2, 6) & = 22 \\ f (0, 7) & = 21 \\ Maximum: f (2, 6) & = 22 \end{matrix}

Step by Step

Question 6

Hard

Find minimum and maximum values of $z = 3 x + y$ ; subject to the constraints: $3 x + 5 y \geq 15, x + 3 y \leq 9, x \geq 0, y \geq 0$

Quick Solution

Under Review

\begin{matrix} 3 x + 5 y & \geq 15, x + 3 y \leq 9, x, y \geq 0 \\ Corner points: & B (0, 3), A (5, 0), C (9, 0) \\ z (0, 3) & = 3 \\ z (5, 0) & = 15 \\ z (9, 0) & = 27 \\ Minimum: z (0, 3) & = 3 \\ Maximum: z (9, 0) & = 27 \end{matrix}

Step by Step

The exercises and question numbering reproduced on these pages are from Mathematics for Class 9 (National Curriculum of Pakistan 2023), published by the Punjab Curriculum and Textbook Board (PCTB), Lahore, authored by Muhammad Akhtar Shirani, Madiha Mahmood, and Ghulam Murtaza. PCTB holds the copyright in the original textbook. PrepSure is not affiliated with, endorsed by, or sponsored by PCTB. The worked solutions, explanations, hints and method notes are PrepSure's own original work, written and reviewed by our team. If you hold rights in this material and believe anything here exceeds fair use, write to us and we will take it down.

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