What is the unique solution of this system?

This still seems ambiguous.

You will have one solution in two linear equations if the slopes are not the same. Given one slope, the other slope can be anything else but.

-6x + 6y = 4
6y = 6x + 4
y = x + 2/3

Slope is 1

So the other slope cannot be 1:

-18x + ky = 16
ky = 18x + 16
y = (18/k)x + 16/k

We don't want the slope to be 1, so:

18/k = 1
18 = k

k can be anything except for 18 to get a value for x and a value for y.

But since k can be anything else, changing k will give you different x's and y's, so I think there is something else missing here if you are expected to get a single answer for k, x, and y.

Also note that k cannot be zero for obvious reasons.

The whole "three unknowns and two equations" thing, you need three equations to solve for three unknowns.