I need to determine a value of K for which the system has infinity many solutions
3x+4y=12
x+ky=4
Then
x=4-(4/3)y
4-(4/3)y+ky=4
-(4/3)y+ky=0
So if k=4/3, then y=y?
If you multiply both sides of the second equation by 3, you'll see
$\displaystyle 3x + 4y = 12$
$\displaystyle 3x + 3ky = 12$.
For the system to have infinitely many solutions, the equations would have to be identically equal.
That means $\displaystyle 3k = 4$ and so $\displaystyle k = \frac{4}{3}$.