What value of k will make the system y-5x=-1 and y=kx +3 inconsistent?

Inconsistent parallel lines is easiest pick k so the slopes (k) is the same as the first line.

It would be helpful to rewrite y-5x=-1 in y=mx+b where m is slope and b is y-intercept

y=5x-1  and y=kx+3  choose k=5 and you will get parallel lines and since the y intercepts are different they will be different parallel lines and inconsistent.

Now  consistent intersecting lines are easy now you know the parallel solution is inconsistent so you can choose k to be what ever you want as long as it isn't 5, to make perpendicular just use -1/5.

Explanation: