A line parallel to $3x-7y = 65$ passes through the point $(7,4)$ and $(0,K)$. What is the value of K?
To find a line parallel to the given line, I need to use the same slope. I can find the slope of the given line by rewriting it in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. To do that, I need to isolate $y$ on one side of the equation. I can do this by adding $7y$ to both sides, then dividing everything by $7$. This gives me $y = \frac{3}{7}x - \frac{65}{7}$. Now I can see that the slope of the given line is $\frac{3}{7}$. So any line parallel to it must also have a slope of $\frac{3}{7}$. I can use this slope and the coordinates of one of the given points, say $(7,4)$, to write an equation for the parallel line in point-slope form, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point. Plugging in the values, I get $y - 4 = \frac{3}{7}(x - 7)$. To find the value of $K$, I need to plug in $x = 0$ and solve for $y$. This gives me $y - 4 = \frac{3}{7}(0 - 7)$, or $y - 4 = -3$. Adding $4$ to both sides, I get $y = 1$. This means that $K = 1$, since $y = K$ when $x = 0$. 