(i) For which values of a and b will the following pair of linear equations have an infinite number of solutions?
2x + 3y = 7
(a - b) x + (a + b) y = 3a + b - 2
(ii) For which value of k will the following pair of linear equations have no solution?
3x + y = 1
(2k -1) x + (k -1) y = 2k +1

Solution:

(i) 2x + 3y - 7 = 0

(a - b) x + (a + b) y - (3a + b - 2) = 0

a₁/a₂ = 2/(a - b)

b₁/b₂ = 3/(a + b)

c₁/c₂ = - 7/[-(3a + b - 2)] = 7/(3a + b - 2)

For infinitely many solutions,

a₁/a₂ = b₁/b₂ = c₁/c₂

2/(a - b) = 7/(3a + b - 2)

6a + 2b - 4 = 7a - 7b

a - 9b = - 4 ....(1)

2/(a - b) = 3/(a + b)

2a + 2b = 3a - 3b

a - 5b = 0 ....(2)

Subtracting (1) from (2), we obtain

(a - 5b) - (a - 9b) = 0 - (-4)

4b = 4

b = 1

Substituting b = 1 in equation (2), we obtain

a - 5 × 1 = 0

a = 5

Hence, a = 5 and b = 1 are the values for which the given equations will have infinitely many solutions.

(ii) 3x + y - 1 = 0

(2k - 1) x + (k -1) y - 2k - 1 = 0

a₁/a₂ = 3/(2k - 1)

b₁/b₂ = 1/(k - 1)

c₁/c₂ = -1/(-2k - 1) = 1/(2k + 1)

For no solution,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

3/(2k - 1) = 1/(k - 1) ≠ 1/(2k + 1)

3/(2k - 1) = 1/(k - 1)

3k - 3 = 2k -1

k = 2

Hence, for k = 2 the given equation has no solution.

☛ Check: Class 10 Maths NCERT Solutions Chapter 3

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Video Solution:

(i) For which values of a and b will the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b) x + (a + b) y = 3a + b - 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k -1) x + (k -1) y = 2k +1

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.5 Question 2

Summary:

(i) The values of a and b for which the equations 2x + 3y = 7 and (a - b) x + (a + b) y = 3a + b - 2 will have infinitely many solutions will be a = 5 and b = 1. (ii) The value of k for wich the equations 3x + y = 1 and (2k -1) x + (k -1) y = 2k + 1 will have no solution is k = 2.

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☛ Related Questions:

• Which of the following pairs of linear equations has a unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using the cross multiplication method. (i) x - 3y - 3 = 0; 3x - 9 y - 2 = 0 (ii) 2x + y = 5; 3x + 2 y = 8 (iii) 3x - 5 y =20; 6x - 10 y = 40 (iv) x - 3y - 7 = 0; 3x - 3y - 15 = 0
• Solve the following pair of linear equations by the substitution and cross-multiplication methods: 8x + 5 y = 9 3x + 2 y = 4.
• Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method: