The reason they give you the perpendicular line is so that you can find the points where that perpendicular line intersects each of the parallel lines, and then use the distance formula between those two points to find the distance between the lines.
`-.5x + 7 = 2x + 7`
`x = 0`
The first point is (0, 7)
`-.5x + 7 = 2x - 3`
`x = 4`
The second point is (4, 5)
Using distance formula
`sqrt((4 - 0)^2 + (5 - 7)^2)`
Actually, to calculate distance between two parallel lines their equations are sufficient.
If we have two lines in the form
ax + by = c1 and ax+by = c2,
then the distance is |c1-c2|/sqrt(a^2+b^2).
Here a=2, b=-1, c1=-7, c2=3. So the answer is |-7-3|/sqrt(2^2+1^2) = 10/sqrt(5) = 2*sqrt(5). It is approx.= 4,472.
if we have to use this perpendicular line, then find the intersections:
1) y=2x+7, y=-0.5x+7.
2x+7=-0.5x+7, 2x=-0.5x, x=0. Then y=7.
2) y=2x-3, y=-0.5x+7.
2x-3=-0.5x+7, 2.5x=10, x=4. Then y=5.
The distance between (0, 7) and (4, 5) is
sqrt(4^2 + 2^2) = sqrt(20) = 2*sqrt(5).