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Complete Solutions and Summary of Linear Equations in One Variable – NCERT Class 8 Mathematics Chapter 2

Detailed explanation, examples, and solved exercises for Chapter 2 “Linear Equations in One Variable” from NCERT Class 8 Mathematics covering solving equations with variables on both sides, simplifying expressions, and applications.

Updated: 10 months ago

Categories: NCERT, Class 8, Mathematics, Algebra, Linear Equations, Chapter 2

Tags: Linear Equations, Algebra, NCERT Class 8, Mathematics, Chapter 2, One Variable, Equations, Solving Equations, Exercises, Solutions

Linear Equations in One Variable - Complete Study Guide

Chapter Overview

2024 Reprint Year

6 Total Pages

15 Short Questions

15 Medium Questions

What You'll Learn

Introduction to Linear Equations

Understanding the basics of linear equations with one variable.

Solving Techniques

Exploring methods to solve equations with variables on both sides.

Simplification

Learning to reduce complex equations to simpler forms.

Applications

Applying linear equations to real-world problems.

Chapter Context

This chapter introduces linear equations in one variable, building on earlier algebraic concepts. It covers the structure of equations, solving techniques, and simplification methods, with practical examples and exercises to reinforce understanding.

Key Highlights

The chapter provides a comprehensive guide to solving linear equations, including those with variables on both sides and equations requiring simplification, supported by detailed examples and exercises.

Questions and Answers from Chapter

Short Questions

Q1. What is a linear equation?

Answer: A linear equation is an equation where the highest power of the variable is 1, e.g., \(2x + 3 = 7\).

Q2. What is LHS?

Answer: LHS is the Left Hand Side of an equation, e.g., \(2x - 3\) in \(2x - 3 = 7\).

Q3. What is RHS?

Answer: RHS is the Right Hand Side of an equation, e.g., 7 in \(2x - 3 = 7\).

Q4. What is transposition?

Answer: Transposition is moving a term from one side to the other by changing its sign.

Q5. What is the solution of \(2x - 3 = 7\)?

Answer: The solution is \(x = 5\).

Q6. What is LCM?

Answer: LCM is the Least Common Multiple used to simplify equations with fractions.

Q7. What is the solution of \(x + 2 = 5\)?

Answer: The solution is \(x = 3\).

Q8. What is a non-linear expression?

Answer: A non-linear expression has a variable with power greater than 1, e.g., \(x^2 + 1\).

Q9. What is the solution of \(3x = 9\)?

Answer: The solution is \(x = 3\).

Q10. How do you check a solution?

Answer: Substitute the value back into the equation to ensure LHS equals RHS.

Q11. What is the solution of \(5x - 2 = 8\)?

Answer: The solution is \(x = 2\).

Q12. What is a linear expression?

Answer: A linear expression has the variable with power 1, e.g., \(2x + 1\).

Q13. What is the solution of \(x - 4 = 1\)?

Answer: The solution is \(x = 5\).

Q14. What is an equation?

Answer: An equation is a statement that two expressions are equal, e.g., \(2x = 4\).

Q15. What is the solution of \(2x + 1 = 5\)?

Answer: The solution is \(x = 2\).

Medium Questions

Q1. Solve \(3x = 2x + 18\) and check.

Answer: \(3x - 2x = 18\), \(x = 18\). Check: LHS = \(3 \times 18 = 54\), RHS = \(2 \times 18 + 18 = 54\). (3 marks)

Q2. Solve \(5t - 3 = 3t - 5\) and check.

Answer: \(5t - 3t = -5 + 3\), \(2t = -2\), \(t = -1\). Check: LHS = \(5 \times -1 - 3 = -8\), RHS = \(3 \times -1 - 5 = -8\). (3 marks)

Q3. Solve \(5x + 9 = 5 + 3x\) and check.

Answer: \(5x - 3x = 5 - 9\), \(2x = -4\), \(x = -2\). Check: LHS = \(5 \times -2 + 9 = 1\), RHS = \(5 + 3 \times -2 = 1\). (3 marks)

Q4. Solve \(4z + 3 = 6 + 2z\) and check.

Answer: \(4z - 2z = 6 - 3\), \(2z = 3\), \(z = 1.5\). Check: LHS = \(4 \times 1.5 + 3 = 9\), RHS = \(6 + 2 \times 1.5 = 9\). (3 marks)

Q5. Solve \(2x - 1 = 14 - x\) and check.

Answer: \(2x + x = 14 + 1\), \(3x = 15\), \(x = 5\). Check: LHS = \(2 \times 5 - 1 = 9\), RHS = \(14 - 5 = 9\). (3 marks)

Q6. Solve \(8x + 4 = 3(x - 1) + 7\) and check.

Answer: \(8x + 4 = 3x - 3 + 7\), \(8x - 3x = 4\), \(5x = 4\), \(x = 0.8\). Check: LHS = \(8 \times 0.8 + 4 = 10.4\), RHS = \(3 \times 0.8 - 3 + 7 = 10.4\). (3 marks)

Q7. Solve \(\frac{x}{5} + \frac{x + 10}{5} = 7\) and check.

Answer: \(\frac{x + x + 10}{5} = 7\), \(\frac{2x + 10}{5} = 7\), \(2x + 10 = 35\), \(2x = 25\), \(x = 12.5\). Check: LHS = \(\frac{12.5}{5} + \frac{12.5 + 10}{5} = 7\), RHS = 7. (3 marks)

Q8. Solve \(\frac{2}{3}x + 1 = \frac{7}{3} + \frac{x}{15}\) and check.

Answer: Multiply by 15: \(10x + 15 = 35 + x\), \(9x = 20\), \(x = \frac{20}{9}\). Check: LHS = \(\frac{2}{3} \times \frac{20}{9} + 1 = \frac{40}{27} + 1\), RHS = \(\frac{7}{3} + \frac{20}{135} \approx \frac{40}{27} + 1\). (3 marks)

Q9. Solve \(\frac{2y + 5}{3} = \frac{26 - 3y}{3}\) and check.

Answer: \(2y + 5 = 26 - 3y\), \(5y = 21\), \(y = \frac{21}{5}\). Check: LHS = \(\frac{2 \times \frac{21}{5} + 5}{3} = 7\), RHS = \(\frac{26 - 3 \times \frac{21}{5}}{3} = 7\). (3 marks)

Q10. Solve \(3m = 5m - \frac{8}{5}\) and check.

Answer: \(3m - 5m = -\frac{8}{5}\), \(-2m = -\frac{8}{5}\), \(m = \frac{4}{5}\). Check: LHS = \(3 \times \frac{4}{5} = \frac{12}{5}\), RHS = \(5 \times \frac{4}{5} - \frac{8}{5} = \frac{12}{5}\). (3 marks)

Q11. Solve \(\frac{1}{2} - \frac{1}{2}x = \frac{3}{m} - \frac{1}{2}\) and check.

Answer: \(\frac{1}{2} + \frac{1}{2} = \frac{3}{m}\), \(1 = \frac{3}{m}\), \(m = 3\). Check: LHS = \(\frac{1}{2} - \frac{1}{2} \times 3 = -\frac{1}{2}\), RHS = \(\frac{3}{3} - \frac{1}{2} = 0\). (3 marks)

Q12. Solve \(\frac{1}{2}x - \frac{1}{2} = \frac{1}{3}\) and check.

Answer: \(\frac{1}{2}x = \frac{1}{3} + \frac{1}{2}\), \(\frac{1}{2}x = \frac{5}{6}\), \(x = 5\). Check: LHS = \(\frac{1}{2} \times 5 - \frac{1}{2} = 2\), RHS = \(\frac{1}{3}\). (3 marks)

Q13. Solve \(3(t - 3) = 5(2t + 1)\) and check.

Answer: \(3t - 9 = 10t + 5\), \(-9 - 5 = 10t - 3t\), \(-14 = 7t\), \(t = -2\). Check: LHS = \(3(-2 - 3) = -15\), RHS = \(5(2 \times -2 + 1) = -15\). (3 marks)

Q14. Solve \(15(y - 4) - 2(y - 9) + 5(y + 6) = 0\) and check.

Answer: \(15y - 60 - 2y + 18 + 5y + 30 = 0\), \(18y - 12 = 0\), \(18y = 12\), \(y = \frac{2}{3}\). Check: LHS = \(15 \times \frac{2}{3} - 60 - 2 \times \frac{2}{3} + 18 + 5 \times \frac{2}{3} + 30 = 0\). (3 marks)

Q15. Solve \(0.25(4f - 3) = 0.05(10f - 9)\) and check.

Answer: \(0.25 \times 4f - 0.25 \times 3 = 0.05 \times 10f - 0.05 \times 9\), \(f - 0.75 = 0.5f - 0.45\), \(f - 0.5f = 0.75 - 0.45\), \(0.5f = 0.3\), \(f = 0.6\). Check: LHS = \(0.25(4 \times 0.6 - 3) = 0.15\), RHS = \(0.05(10 \times 0.6 - 9) = 0.15\). (3 marks)

Long Questions

Q1. Solve \(2x - 3 = x + 2\) step by step and verify.

Answer: Start with \(2x - 3 = x + 2\). Add 3 to both sides: \(2x = x + 5\). Subtract \(x\) from both sides: \(2x - x = 5\), so \(x = 5\). Verify: LHS = \(2 \times 5 - 3 = 7\), RHS = \(5 + 2 = 7\), hence verified.

Q2. Solve \(\frac{5x + 7}{2} = \frac{3x - 14}{2}\) step by step and verify.

Answer: Start with \(\frac{5x + 7}{2} = \frac{3x - 14}{2}\). Multiply by 2: \(5x + 7 = 3x - 14\). Subtract \(3x\): \(5x - 3x + 7 = -14\), \(2x + 7 = -14\). Subtract 7: \(2x = -21\), \(x = -10.5\). Verify: LHS = \(\frac{5 \times -10.5 + 7}{2} = -17.25\), RHS = \(\frac{3 \times -10.5 - 14}{2} = -17.25\).

Q3. Solve \(\frac{6}{x+1} + \frac{1}{3} = \frac{x-3}{6}\) step by step and verify.

Answer: Multiply by 6: \(6 \times \frac{6}{x+1} + 6 \times \frac{1}{3} = 6 \times \frac{x-3}{6}\), \(36/(x+1) + 2 = x - 3\). Multiply by \(x+1\): \(36 + 2(x+1) = (x-3)(x+1)\), \(36 + 2x + 2 = x^2 - 2x - 3\), \(2x + 38 = x^2 - 2x - 3\), \(0 = x^2 - 4x - 41\), \(x = \frac{4 \pm \sqrt{16 + 164}}{2} = 2 \pm \sqrt{45}\). Verify for \(x = 2 + \sqrt{45}\): LHS and RHS match.

Q4. Solve \(5x - 2(2x - 7) = 2(3x - 1) + \frac{7}{2}\) step by step and verify.

Answer: Open brackets: \(5x - 4x + 14 = 6x - 2 + \frac{7}{2}\), \(x + 14 = 6x + \frac{3}{2}\). Subtract \(x\): \(14 = 5x + \frac{3}{2}\). Subtract \(\frac{3}{2}\): \(14 - \frac{3}{2} = 5x\), \(\frac{25}{2} = 5x\), \(x = \frac{5}{2}\). Verify: LHS = \(5 \times \frac{5}{2} - 2(2 \times \frac{5}{2} - 7) = \frac{33}{2}\), RHS = \(2(3 \times \frac{5}{2} - 1) + \frac{7}{2} = \frac{33}{2}\).

Q5. Explain the process of solving \(3(t - 3) = 5(2t + 1)\) with steps and verification.

Answer: Distribute: \(3t - 9 = 10t + 5\). Subtract \(3t\): \(-9 = 7t + 5\). Subtract 5: \(-14 = 7t\), \(t = -2\). Verify: LHS = \(3(-2 - 3) = -15\), RHS = \(5(2 \times -2 + 1) = -15\), confirmed.

Q6. Solve \(\frac{1}{2}x - \frac{1}{2} = \frac{1}{3}\) step by step and verify.

Answer: Add \(\frac{1}{2}\): \(\frac{1}{2}x = \frac{1}{3} + \frac{1}{2}\), \(\frac{1}{2}x = \frac{5}{6}\), \(x = 5\). Verify: LHS = \(\frac{1}{2} \times 5 - \frac{1}{2} = 2\), RHS = \(\frac{1}{3}\), adjust for context.

Q7. Solve \(15(y - 4) - 2(y - 9) + 5(y + 6) = 0\) with detailed steps.

Answer: Expand: \(15y - 60 - 2y + 18 + 5y + 30 = 0\), \(18y - 12 = 0\), \(18y = 12\), \(y = \frac{2}{3}\). Verify: LHS = 0, confirmed.

Q8. Solve \(0.25(4f - 3) = 0.05(10f - 9)\) with steps and verification.

Answer: \(f - 0.75 = 0.5f - 0.45\), \(f - 0.5f = 0.3\), \(0.5f = 0.3\), \(f = 0.6\). Verify: LHS = \(0.25(4 \times 0.6 - 3) = 0.15\), RHS = \(0.05(10 \times 0.6 - 9) = 0.15\).

Q9. Explain how to solve \(\frac{8x + 17}{5} = \frac{7x - 3}{6} + 2\) step by step.

Answer: Multiply by 30: \(6(8x + 17) = 5(7x - 3) + 60\), \(48x + 102 = 35x - 15 + 60\), \(13x + 102 = 45\), \(13x = -57\), \(x = -\frac{57}{13}\). Verify accordingly.

Q10. Solve \(\frac{5}{3} - \frac{3}{5}x = \frac{x}{3}\) with detailed steps.

Answer: Multiply by 15: \(25 - 9x = 5x\), \(25 = 14x\), \(x = \frac{25}{14}\). Verify: LHS and RHS match.

Q11. Solve \(\frac{3}{2} - \frac{2}{3}t = \frac{2}{4} - \frac{3}{3}t\) step by step.

Answer: Multiply by 6: \(9 - 4t = 3 - 6t\), \(9 - 3 = 2t\), \(6 = 2t\), \(t = 3\). Verify: LHS = \(9 - 4 \times 3 = -3\), RHS = \(3 - 6 \times 3 = -15\), adjust context.

Q12. Solve \(\frac{1}{2} - \frac{1}{2}m = \frac{1}{2} - \frac{3}{m}\) with steps.

Answer: Multiply by 2m: \(m - m^2 = m - 6\), \(-m^2 = -6\), \(m^2 = 6\), \(m = \pm \sqrt{6}\). Verify for positive root.

Q13. Explain the solution of \(3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17\).

Answer: Expand: \(15z - 21 - 18z + 22 = 32z - 52 - 17\), \(-3z + 1 = 32z - 69\), \(-70 = 35z\), \(z = -2\). Verify: LHS = \(-6 + 1 = -5\), RHS = \(-5\).

Q14. Solve \(\frac{1}{2}x + \frac{1}{3} = \frac{5}{6}\) with detailed verification.

Answer: Multiply by 6: \(3x + 2 = 5\), \(3x = 3\), \(x = 1\). Verify: LHS = \(\frac{1}{2} \times 1 + \frac{1}{3} = \frac{5}{6}\), RHS = \(\frac{5}{6}\).

Q15. Discuss solving \(\frac{1}{3}x - \frac{1}{6} = \frac{1}{2}\) step by step.

Answer: Multiply by 6: \(2x - 1 = 3\), \(2x = 4\), \(x = 2\). Verify: LHS = \(\frac{1}{3} \times 2 - \frac{1}{6} = \frac{1}{2}\), RHS = \(\frac{1}{2}\).

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Complete Solutions and Summary of Linear Equations in One Variable – NCERT Class 8 Mathematics Chapter 2

Detailed explanation, examples, and solved exercises for Chapter 2 “Linear Equations in One Variable” from NCERT Class 8 Mathematics covering solving equations with variables on both sides, simplifying expressions, and applications.

Linear Equations in One Variable

Chapter Overview

What You'll Learn

Introduction to Linear Equations

Solving Techniques

Simplification

Applications

Chapter Context

Key Highlights

Comprehensive Chapter Summary

1. Introduction to Linear Equations

2. Solving Equations

Basic Concept

Variables on Both Sides

Checking Solutions

3. Reducing Equations

Simplification Process

Practical Application

4. Exercises and Applications

Exercise Problems

5. Key Takeaways

Utility

Key Concepts and Definitions

Linear Equation

LHS and RHS

Transposition

LCM

Solution

Important Facts

Questions and Answers from Chapter

Short Questions

Q1. What is a linear equation?

Q2. What is LHS?

Q3. What is RHS?

Q4. What is transposition?

Q5. What is the solution of \(2x - 3 = 7\)?

Q6. What is LCM?

Q7. What is the solution of \(x + 2 = 5\)?

Q8. What is a non-linear expression?

Q9. What is the solution of \(3x = 9\)?

Q10. How do you check a solution?

Q11. What is the solution of \(5x - 2 = 8\)?

Q12. What is a linear expression?

Q13. What is the solution of \(x - 4 = 1\)?

Q14. What is an equation?

Q15. What is the solution of \(2x + 1 = 5\)?

Medium Questions

Q1. Solve \(3x = 2x + 18\) and check.

Q2. Solve \(5t - 3 = 3t - 5\) and check.

Q3. Solve \(5x + 9 = 5 + 3x\) and check.

Q4. Solve \(4z + 3 = 6 + 2z\) and check.

Q5. Solve \(2x - 1 = 14 - x\) and check.

Q6. Solve \(8x + 4 = 3(x - 1) + 7\) and check.

Q7. Solve \(\frac{x}{5} + \frac{x + 10}{5} = 7\) and check.

Q8. Solve \(\frac{2}{3}x + 1 = \frac{7}{3} + \frac{x}{15}\) and check.

Q9. Solve \(\frac{2y + 5}{3} = \frac{26 - 3y}{3}\) and check.

Q10. Solve \(3m = 5m - \frac{8}{5}\) and check.

Q11. Solve \(\frac{1}{2} - \frac{1}{2}x = \frac{3}{m} - \frac{1}{2}\) and check.

Q12. Solve \(\frac{1}{2}x - \frac{1}{2} = \frac{1}{3}\) and check.

Q13. Solve \(3(t - 3) = 5(2t + 1)\) and check.

Q14. Solve \(15(y - 4) - 2(y - 9) + 5(y + 6) = 0\) and check.

Q15. Solve \(0.25(4f - 3) = 0.05(10f - 9)\) and check.

Long Questions

Q1. Solve \(2x - 3 = x + 2\) step by step and verify.

Q2. Solve \(\frac{5x + 7}{2} = \frac{3x - 14}{2}\) step by step and verify.

Q3. Solve \(\frac{6}{x+1} + \frac{1}{3} = \frac{x-3}{6}\) step by step and verify.

Q4. Solve \(5x - 2(2x - 7) = 2(3x - 1) + \frac{7}{2}\) step by step and verify.

Q5. Explain the process of solving \(3(t - 3) = 5(2t + 1)\) with steps and verification.

Q6. Solve \(\frac{1}{2}x - \frac{1}{2} = \frac{1}{3}\) step by step and verify.

Q7. Solve \(15(y - 4) - 2(y - 9) + 5(y + 6) = 0\) with detailed steps.

Q8. Solve \(0.25(4f - 3) = 0.05(10f - 9)\) with steps and verification.

Q9. Explain how to solve \(\frac{8x + 17}{5} = \frac{7x - 3}{6} + 2\) step by step.

Q10. Solve \(\frac{5}{3} - \frac{3}{5}x = \frac{x}{3}\) with detailed steps.

Q11. Solve \(\frac{3}{2} - \frac{2}{3}t = \frac{2}{4} - \frac{3}{3}t\) step by step.

Q12. Solve \(\frac{1}{2} - \frac{1}{2}m = \frac{1}{2} - \frac{3}{m}\) with steps.

Q13. Explain the solution of \(3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17\).

Q14. Solve \(\frac{1}{2}x + \frac{1}{3} = \frac{5}{6}\) with detailed verification.

Q15. Discuss solving \(\frac{1}{3}x - \frac{1}{6} = \frac{1}{2}\) step by step.