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 = \(\frac{1}{2}\).
