Question 107501
<pre><font size = 4><b>
<font color = "green">y = 4x</font> 
y = 4x - 17
<font color = "blue">y = 4x - 17</font> 
find the distance between those two parallel lines

First we graph <font color = "green">y = 4x</font> in green

{{{graph(300,700,-8,8,-20,20,0,4x)}}}

Now on the same axes, we graph <font color = "blue">y = 4x - 17</font> in blue:

{{{drawing(300,700,-8,8,-20,20,
   graph(300,700,-8,8,-20,20,0,4x,4x-17)
)}}}

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:

{{{drawing(300,700,-8,8,-20,20,
   graph(300,700,-8,8,-20,20,0,4x,4x-17,0,0,0, -.25x)
)}}}

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

<font color = "green">y = 4x</font>
 
We can compare that to

y = mx + b

by writing <font color = "green">y = 4x</font> as

<font color = "green">y = 4x + 0</font>

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:

<font color = "blue">y = 4x - 17</font>

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 {{{-1/4}}}.  So 
the purple line has slope m = {{{-1/4}}}.  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 {{{-1/4}}} for m and 0 for b into

y = mx + b 

<font color = "indigo">y = {{{-1/4}}} + 0 x</font>

or just

<font color = "indigo">y = {{{-1/4}}}x</font>

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

<font color = "indigo">y = {{{-1/4}}}x</font>
<font color = "blue">y = 4x - 17</font> 
 
I assume you can solve those.  You get the
point (x, y) = (4, -1)

Now we mark the point (4, -1)

{{{drawing(300,700,-8,8,-20,20,
   graph(300,700,-8,8,-20,20,0,4x,4x-17,0,0,0, -.25x),
   locate(4,-1.2,"(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 = <font face = "symbol">Ö</font>(x<sub>2</sub> - x<sub>1</sub>)² + (y<sub>2</sub> - y<sub>1</sub>)²

where (x<sub>1</sub>, y<sub>1</sub>) = (0, 0) and (x<sub>2</sub>, y<sub>2</sub>) = (4, -1)

     ____________________
d = <font face = "symbol">Ö</font>(4 - 0)² + (-1 - 0)²
     ____________
d = <font face = "symbol">Ö</font>(4)² + (-1)²
     ______
d = <font face = "symbol">Ö</font>16 + 1
     __
d = <font face = "symbol">Ö</font>17

Edwin</pre>