Question 213539
Find the distance between the lines with equations 3x - y = 9 and y = 3x - 4.

a)5/4
b)5 sqrt(10)/2
c) sqrt(10)/2
d) 13 sqrt(10)/2


Step 1.  Here's a graph of the two equations:


{{{graph( 300, 400, -5, 5, -10, 10, y=3x-9, y=3x-4)}}}


Step 2.  We need a line that's perpendicular to the two parallel lines and then use the point where it intercepts the axis.  The slope of the above two lines is 3 and the perpendicular line has a slope of -1/3.  The reason being is the fact that the product of the slope for two perpendicular lines is equal to -1.


Step 3.  So let's pass the perpendicular line  through (0,-4) and then we need to solve y=3x-4 line and figure out where it intersects the the other parallel line.


So our equation in slope intercept form is  y=mx+b=-4=-0/3+b. So b=-4.  The the equation of the perpendicular line is:


{{{y=-1*x/3+4/3}}}

Here a graph of the situation


{{{graph( 300, 400, -5, 5, -10, 10, y=3x-4, y=3x-9, y=(-x/3)-4)}}}


Step 4.  For visually simplicity, let's pick a point say (0,-4) as one point. We can find the other point, using a system of equations


{{{y=(-x/3)-4}}}


{{{y=3x-9}}}


The following steps will solve this system of equations



*[invoke linear_substitution "x", "y", 1/3, 1, -4, -3, 1, -9 ]


Step 5.  Based on the above the other point (1.5 and -4.5)


Step 6.  The distance between (0,-4) and (1.5, -4.5) is therefore,  


{{{d=sqrt((0-1.5)^2+(-4-(-4.5)))}}}


{{{d=sqrt(2.25+0.25)}}}


{{{d=sqrt(2.5)=sqrt(10/4))}}}


{{{d=sqrt(10)/2}}}


{{{d=1.58}}}


Step 7.  The answer is c or 1.58.



I hope the above steps were helpful and it's probably long.  I'll see if there is a shorter way.  But that's my approach  for now.


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Good luck in your studies!


Respectfully,
Dr J