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:

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