jmc

We put the equation of the line into slope-intercept form by solving for $y$: $y=\frac{65-3x}{-7}$. That means the slope of the line is $\frac{3}{7}$, and the slope of a parallel line must also be $\frac{3}{7}$. The slope of the line through $(7,4)$ and $(0,K)$ is $\frac{4-K}{7-0}$, which we set equal to $\frac{3}{7}$ and solve for $K$. $$\frac{4-K}{7}=\frac{3}{7}\Rightarrow 4-K=3\Rightarrow 1=K$$ So the value of K is $\boxed{1}$.