Explanation:

KX + 3Y - K + 3 = 0 --------(1)
12X + KY = K
12X + KY - K = 0 ------(2)
These equations are of the form of A1X+ B1Y + C1 = 0 and A2X + B2Y + C2 = 0
Where,
A1 = K , B1 = 3 and C2 = -K+3
And,
A2 = 12 , B2 = K and C2 = -K
For infinite solution we must have :
A1/A2 = B1/B2 = C1/C2
K/12 = 3/K = -K+3/-K
K/12 = 3/K = K-3/K
K/12 = 3/K and 3/K = K-3/K
K² = 36 and K²-6K = 0
K = ?36 and K ( K -6 ) = 0
K = 6 or -6 and ( K = 0 or K = 6 )
K = 6
Hence, The given equations has infinitely many solution when k = 6.