y = 4x 
y = 4x - 17
y = 4x - 17 
find the distance between those two parallel lines

First we graph y = 4x in green



Now on the same axes, we graph y = 4x - 17 in blue:



Now we need to find the equation of a line through the 
origin (0,0) which is perpendicular to both of them.  
That is, we need to find the equation of the purple
line below:



Let's find the slope and y-intercepts of the green
and blue lines:

y = 4x
 
We can compare that to

y = mx + b

by writing y = 4x as

y = 4x + 0

and we find that the slope m = 4 and b = 0,
so the slope of the green line is 4 and its
y-intercept is (0,b) which is (0,0) since b = 0.

Now the blue line:

y = 4x - 17

We can compare that to

y = mx + b

and we find that the slope m = 4 and b = -17,
so the slope of the blue line is 4 and its
y-intercept is (0,b) which is (0,-17) since b = -17. 

Since the slopes of both the green and blue lines are 
both 4, we know that the slope of any line perpendicular 
to them must have a slope which is the reciprocal of 4 
with its sign changed.  That would be .  So 
the purple line has slope m = .  Since the
purple line goes through the origin (0,0), and
since the origin is on the y-axis, then b = 0.
So the purple line's equation can be found from
substituting  for m and 0 for b into

y = mx + b 

y =  + 0 x

or just

y = x

Now we need to find the point where the purple
line crosses the blue line. So we solve this
system of equations:

y = x
y = 4x - 17 
 
I assume you can solve those.  You get the
point (x, y) = (4, -1)

Now we mark the point (4, -1)




So all we need do now is find the distance
between the origin (0, 0) and the point (4, -1)

We use the distance formula:
     _______________________
d = Ö(x2 - x1)² + (y2 - y1)²

where (x1, y1) = (0, 0) and (x2, y2) = (4, -1)

     ____________________
d = Ö(4 - 0)² + (-1 - 0)²
     ____________
d = Ö(4)² + (-1)²
     ______
d = Ö16 + 1
     __
d = Ö17

Edwin