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Find the value of k for which the system of equations x + 2y = 3 and (k − 1)x + (k + 1)y = k + 3 represents a pair of coincident lines.
Coincident lines represent identical equations requiring equal coefficient ratios: a₁/a₂ = b₁/b₂ = c₁/c₂. Setting up the ratios gives 1/(k − 1) = 2/(k + 1) = 3/(k + 3). Equating the first two ratios yields k + 1 = 2(k − 1), which simplifies to k = 3. Substituting k = 3 into all three ratios yields 1Read more
Coincident lines represent identical equations requiring equal coefficient ratios: a₁/a₂ = b₁/b₂ = c₁/c₂. Setting up the ratios gives 1/(k − 1) = 2/(k + 1) = 3/(k + 3). Equating the first two ratios yields k + 1 = 2(k − 1), which simplifies to k = 3. Substituting k = 3 into all three ratios yields 1/2 = 2/4 = 3/6 = 1/2, confirming k = 3.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 13 Two Variables, One Line Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-13/
See lessFind the values of a and b for which the system of equations (a + b)x − 2by = 5a + 2b + 1 and 3x − y = 14 has infinitely many solutions.
For a system to possess infinitely many solutions, the ratio of coefficients must satisfy a₁/a₂ = b₁/b₂ = c₁/c₂. Comparing gives (a + b)/3 = 2b = (5a + 2b + 1)/14. From (a + b)/3 = 2b, we get a = 5b. From (5a + 2b + 1)/14 = 2b, we get 5a − 26b = −1. Substituting a = 5b gives 25b − 26b = −1, yieldingRead more
For a system to possess infinitely many solutions, the ratio of coefficients must satisfy a₁/a₂ = b₁/b₂ = c₁/c₂. Comparing gives (a + b)/3 = 2b = (5a + 2b + 1)/14. From (a + b)/3 = 2b, we get a = 5b. From (5a + 2b + 1)/14 = 2b, we get 5a − 26b = −1. Substituting a = 5b gives 25b − 26b = −1, yielding b = 1 and a = 5.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 13 Two Variables, One Line Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-13/
See lessFor what values of p does the pair of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 have a unique solution?
A pair of linear equations has a unique solution if and only if the ratio of the coefficients of x and y are not equal, that is, a₁/a₂ ≠ b₁/b₂. From the equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0, this requires 4/2 ≠ p/2. Simplifying yields 2 ≠ p/2, which means p ≠ 4. Therefore, the system has aRead more
A pair of linear equations has a unique solution if and only if the ratio of the coefficients of x and y are not equal, that is, a₁/a₂ ≠ b₁/b₂. From the equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0, this requires 4/2 ≠ p/2. Simplifying yields 2 ≠ p/2, which means p ≠ 4. Therefore, the system has a unique solution for all real values of p except 4.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 13 Two Variables, One Line Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-13/
See lessHow many lines exist that have a given slope and pass through a given point?
Only one unique line can exist with a given slope and passing through a given point. According to the point-slope form y − y₁ = m(x − x₁), fixing both the slope m and the coordinate point (x₁, y₁) uniquely establishes the intercept c = y₁ − mx₁. Since no other line can share both characteristics, thRead more
Only one unique line can exist with a given slope and passing through a given point. According to the point-slope form y − y₁ = m(x − x₁), fixing both the slope m and the coordinate point (x₁, y₁) uniquely establishes the intercept c = y₁ − mx₁. Since no other line can share both characteristics, the line is unique.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 13 Two Variables, One Line Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-13/
See lessHow many lines exist that have a given slope?
Infinitely many lines exist that share a given slope. Any straight line with slope m can be represented by the equation y = mx + c. Because the y-intercept parameter c can assume any real number value, each distinct value of c defines a unique line parallel to the others, resulting in an infinite faRead more
Infinitely many lines exist that share a given slope. Any straight line with slope m can be represented by the equation y = mx + c. Because the y-intercept parameter c can assume any real number value, each distinct value of c defines a unique line parallel to the others, resulting in an infinite family of parallel lines having identical slope.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 13 Two Variables, One Line Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-13/
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