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
