Question

Find the distance between parallel lines y=5x+2 and y=5x-7.

Answer 1

First determine a convinient point on either line!
Let us say we have the first one
And the point can be (1,7) [BECAUSE IT FITS THE EQUATION]
Okay so that being said we move on to our perpendicular!
Now you should know that the slopes of Perpendicular lines are related as
m1.m2 = -1
And Parrallel lines obviously always have the same slope!

Answer 2

So the slope of the parrallel lines in these equations are -1/5
Okay so now we have a point (1,7) and slope of perpendicular as -1/5
Use equation = (y-y1) = m(x-x1) to find the perpendicular equation!!!

Answer 3

After you are done with that use the perpendicular line and the other parrallel line to find their solution! [YOU GET THEIR INTERSECTION]
After you get the two points where the perpendicular intersects with the 2 parrallel lines
Use the distance formula [DISTANCE BETWEEN 2 POINTS]
To get the shortest distance/Perpendicular between the 2 parralel lines!
Understood?

Answer 4

Refer to this diagram.

Answer 5

Understand? :D

Answer 6

Thank you!

Answer 7

would the distance be 4?

Answer 8

Another way to go about this is by using the distance formula directly:
\[d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\]
Here, \((x_1,y_1)\) is a point on one line (\(y=5x+2\)), and \((x_2,y_2)\) is a point on the other line (\(y=5x-7\)).

So your first point is equivalent to \((x_1,5x_1+2)\) and your second point is \((x_2,5x_2-7)\):
\[d=\sqrt{(x_1-x_2)^2+(5x_1+2-(5x_2-7))^2}\\
d=\sqrt{(x_1-x_2)^2+(5x_1-5x_2+9)^2}\]
Then it's only a matter of determining the x-coordinates of the intersections of the parallel lines and a perpendicular, like @AkashdeepDeb explained.