Question

Find the values of a and b for which the following system of equations has infinitely many solutions:

(i) (2a - 1)x - 3y = 5

3x + (b - 2)y = 3

(ii) 2x - (2a + 5y)y = 5

(2b + 1)x  -9y  = 15

(iii) (a - 1)x + 3y = 2

6x + (1 - 2b)y = 6

(iv) 3x + 4y = 12

(a + b)x + 2(a - b)y = 5a - 1

(v) 2x + 3y = 7

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

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

(a - 1)x + (a + 1)y = (3a - 1) 

(vii) 2x + 3y = 7

(a - 1)x + (a + 2)y = 3a

Answer 1

(i) For infinitely many solution

⇒ 6a – 3 = 15 and - 9 = 5b – 10

⇒ a = 3 and b = 1/5

(ii) For infinitely many solution

⇒ 6 = 2b + 1 and 6a + 15 = 9

⇒ b = 5/2 and a = -1

(iii) For infinitely many solution

⇒ 3a – 3 = 6 and 9 = 1 – 2b

⇒ a = 3 and b = - 4

(iv) For infinitely many solution

⇒ 15a – 3 = 12a + 12b and 20a – 4 = 24a – 24b

⇒ 3a – 12b = 3 ------ (1)

and 6b – a = 1 ------ (2)

Multiplying eq2 by 3 and adding to eq1

⇒ 6b = 6

⇒ b = 1

Thus,

3a – 12 = 3

⇒ a = 5

(v) For infinitely many solution

⇒ 2a + 2b = 3a – 3b and 6a + 2b – 4 = 7a – 7b

⇒ a = 5b and 9b – a = 4

Thus,

9b – 5b = 4

⇒ b = 1

⇒ a = 5

(vi) For infinitely many solution

⇒ 2a + 2 = 3a – 3

⇒ a = 5

(vii) For infinitely many solution

⇒ 2a + 4 = 3a – 3

⇒ a = 7