SOLUTION: find the distance between the lines with equations y=3x-9 and y = 3x - 4. a)5/4 b)5 sqrt(10)/2 c)sqrt(10)/2 d)13 sqrt(10)/2

Algebra ->  Trigonometry-basics -> SOLUTION: find the distance between the lines with equations y=3x-9 and y = 3x - 4. a)5/4 b)5 sqrt(10)/2 c)sqrt(10)/2 d)13 sqrt(10)/2      Log On


   

Question 213539: find the distance between the lines with equations y=3x-9 and y = 3x - 4.
a)5/4
b)5 sqrt(10)/2
c)sqrt(10)/2
d)13 sqrt(10)/2
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
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%28+300%2C+400%2C+-5%2C+5%2C+-10%2C+10%2C+y=3x-9%2C+y=3x-4%29

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%2Ax%2F3%2B4%2F3
Here a graph of the situation

graph%28+300%2C+400%2C+-5%2C+5%2C+-10%2C+10%2C+y=3x-4%2C+y=3x-9%2C+y=%28-x%2F3%29-4%29

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=%28-x%2F3%29-4

y=3x-9

The following steps will solve this system of equations


Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
We'll use substitution. After moving 1*y to the right, we get:
0.333333333333333%2Ax+=+-4+-+1%2Ay, or x+=+-4%2F0.333333333333333+-+1%2Ay%2F0.333333333333333. Substitute that
into another equation:
-3%2A%28-4%2F0.333333333333333+-+1%2Ay%2F0.333333333333333%29+%2B+1%5Cy+=+-9 and simplify: So, we know that y=-4.5. Since x+=+-4%2F0.333333333333333+-+1%2Ay%2F0.333333333333333, x=1.5.

Answer: system%28+x=1.5%2C+y=-4.5+%29.



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%28%280-1.5%29%5E2%2B%28-4-%28-4.5%29%29%29

d=sqrt%282.25%2B0.25%29

d=sqrt%282.5%29=sqrt%2810%2F4%29%29

d=sqrt%2810%29%2F2

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.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J