Question

Find the values of a and b for which the given system of equations has infinitely many solutions.
2x +3y=9.
(a+b)x(2a-b)y = 3(a+b+1).

Answer 1

L₁ ≡ 2x + 3y = 9
L₂ ≡ (a+b)x + (2a-b)y = 3(a+b+1)

For two lines to have infinitely   solutions , they would have to be coincident lines ,⇒

⇒ratio of coefficients of x = ratio of coefficients of y = ratio of constants
 ⇒2/(a+b) = 3 / (2a-b) = 9/3(a+b+1)
This fetches two equations

⇒ 2(2a-b) = 3 (a+b) ⇒ 4a-2b = 3a + 3b ⇒ a = 5b

and ⇒ 6(a+b+1) = 9(a+b)  ⇒ 6a +6b +6 = 9a +9b ⇒ 3a + 3b = 6
⇒ a+b = 2 ⇒ 5b + b = 2 ⇒ 6b = 2 ⇒ b =1/3   and a = 5/3

Ans a = 5/3 and b = 1/3

Hope my answer is correct.