jmc

A system of linear equations won't have a solution if the two equations in the system contradict each other. Multiplying the first equation by 2, we get the equations $$\begin{array}{rl}12x+8y& =14,\\ Kx+8y& =7.\end{array}$$If $K=12,$ then the two equations contradict each other. Otherwise, we can subtract them to get $(K-12)x=-7.$ We can solve this equation to find $x,$ then substitute to find $y.$

So the value of $K$ for which the system has no solution is $K=\boxed{12}.$