20 less than eight times a number is the same as 15 more than the number

That's all very interesting, but you haven't said WHY you're telling us all of that.  

I have a hunch, and I'm going with my gut ... Unless I miss my guess. I suppose now you want us to tell you what "the number" actually is.  OK. Watch closely.

We don't know what the number is, but in order to work with it, we need something to CALL IT.  I'll call it " Q " .

-- the number . . . Q

-- eight times the number . . . 8Q

-- 20 less than eight times the number . . . 8Q - 20

-- 15 more than the number . . . Q + 15

So you just said that  (8Q - 20) = (Q + 15) , and now we can 'solve' this for Q .

Start with  (8Q - 20) = (Q + 15)

Take that, and subtract Q from each side:

7Q - 20 = 15

Take that, and add 20 to each side:

7Q = 35

Take that, and divide each side by  7 :

Q = 5  <== There's your mystery number.

step-by-step explanation:

points a and b are equidistant from the desired line, so the line you find by using them will go through point r.

using points other than a and b will give the desired line only if they are equidistant from r. if they are randomly located, the perpendicular line you create will not necessarily go through point r, hence will not be a solution to the problem.