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.
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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
Chapter 2: Mathematics
Complete Study Guide with Interactive Learning
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.
Comprehensive Chapter Summary
1. Introduction to Linear Equations
The chapter begins with an introduction to algebraic expressions and equations, focusing on linear equations in one variable where the highest power of the variable is 1, such as \(2x + 1\) and \(3y - 7\).
2. Solving Equations
Basic Concept
An equation balances LHS and RHS, solved by performing the same operations on both sides, e.g., \(2x - 3 = 7\) where \(x = 5\).
Variables on Both Sides
Equations like \(2x - 3 = x + 2\) are solved by transposing variables, resulting in \(x = 5\).
Checking Solutions
Solutions are verified by substituting the value back, ensuring LHS equals RHS.
3. Reducing Equations
Simplification Process
Complex equations like \(\frac{6}{x+1} + \frac{1}{3} = \frac{x-3}{6}\) are simplified by multiplying by the LCM (6), leading to \(x = -1\).
Practical Application
Equations are reduced by opening brackets and combining like terms, as in \(5x - 2(2x - 7) = 2(3x - 1) + \frac{7}{2}\), solved to \(x = \frac{5}{2}\).
4. Exercises and Applications
Exercise Problems
Includes exercises like \(3x = 2x + 18\) and \(3(t - 3) = 5(2t + 1)\) to practice solving techniques.
5. Key Takeaways
Utility
Linear equations are useful for solving problems on numbers, ages, and perimeters.
Key Concepts and Definitions
Linear Equation
An equation where the highest power of the variable is 1, e.g., \(2x + 3 = 7\).
LHS and RHS
Left Hand Side and Right Hand Side of an equation, balanced in a solution.
Transposition
Moving terms from one side to the other by changing their sign.
LCM
Least Common Multiple used to simplify equations with fractions.
Solution
The value of the variable that satisfies the equation.
Important Facts
1
Highest Power in Linear Equations
2024
Reprint Year
6
Total Pages
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 = \(\frac{1}{2}\).
Interactive Knowledge Quiz
Test your understanding of Linear Equations in One Variable
Quick Revision Notes
Basics
- Linear equation: \(ax + b = 0\)
- Power of variable = 1
- Examples: \(2x + 3 = 7\)
Solving
- Transpose terms
- Use LCM for fractions
- Check with substitution
Simplification
- Open brackets
- Combine like terms
- Multiply by LCM
Applications
- Numbers
- Ages
- Perimeters
Exam Strategy Tips
- Practice transposition
- Master LCM usage
- Verify all solutions
- Understand examples
- Solve exercises
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