Question

Find the value of k for which the system of linear equations has an infinite number of solutions: 

kx + 3y = (2k + 1), 

2(k + 1)x + 9y = (7k + 1).

Answer 1

The given system of equations: 

kx + 3y = (2k + 1) 

⇒ kx + 3y - (2k + 1) = 0 ….(i)

And, 2(k + 1)x + 9y = (7k + 1) 

⇒ 2(k + 1)x + 9y - (7k + 1) = 0 …(ii) 

These equations are of the following form: 

a₁x+b₁y+c₁ = 0, a₂x+b₂y+c₂ = 0 

where, a₁ = k, b₁= 3, c₁ = -(2k + 1) and a₂ = 2(k + 1), b₂ = 9, c₂ = -(7k + 1) 

For an infinite number of solutions, we must have:

Now, we have the following three cases: 

Case I:

⇒ k(7k + 1) = (2k + 1) × 2(k + 1) 

⇒ 7k² + k = (2k + 1) (2k + 2) 

⇒ 7k² + k = 4k2 + 4k + 2k + 2 

⇒ 3k² – 5k – 2 = 0 

⇒ 3k² – 6k + k – 2 = 0 

⇒ 3k(k – 2) + 1(k – 2) = 0 

⇒ (3k + 1) (k – 2) = 0

⇒ k = 2 or k = −1/3 

Hence, the given system of equations has an infinite number of solutions when k is equal to 2.