2770.51 – Moving Line

Given two lines, $f(x)=3x+1$ and $g(x)=5-2x$ as well as a third line $L_1:y=mx+c$ with $m>0$ and $c$ a constant. $L_1$ moves parallel to itself and intersects the two given lines at two points: $P_1$ and $P_2$. The locus of midpoints of line segment $P_1 P_2$ is line $L_2$. Find the slope of $L_1$ so that the slope of $L_2$ is undefined.

1. 1/3
2. 1/2
3. 2/3
4. 1
5. 3/2

To put it another way, find the slope of $L_1$ so that, as it moves up and down while maintaining the same slope, the midpoint of the line segment formed by connecting the two places where $L_1$ intersects $f(x)$ and $g(x)$ will always have the same $x$-value.