We need to find the line that is perpendicular to both these parallel lines. Perpendicular lines have negative reciprocal slope.
m = -7/2
So your perpendicular line is now y = (-7/2)x + 4
This line is perpdicular to the first line. The point of intersection is (0, 4). You will use this point later.
Now we set this line equal to the last line to find the point of intersection.
-(7 / 2)x + 4 = (2 / 7)x - 2
Solve for x from this equation.
Multiply both sides of the equation by 2.
-7x + 8 = (4 / 7)x - 4
Multiply both sides of the equation by 7.
-49x + 56 = 4x - 28
-49x - 4x = -28 - 56
-53x = -84
x2 = 1.585
Plug in this value of x into any side to solve for y.
y2 = (2 / 7)(1.585) - 2
This is your second point of intersection: (x2 , y2).
Finally, find the distance between the two points of intersection using the distance formula. That will be your distance between the two given lines.