Playing with Equations to Solve Problems

Date: 09/16/2003 at 10:48:13
From: Mara
Subject: To state the geometric property of an equation

I need to give the geometric property common to all lines in the 
family

   x - ky = 1

I know that the answer is that all lines in this family have an
x-intercept at x=1 but I am totally clueless about showing why this is
the case. 

At first I thought that using the double-intercept equation   
  
  (x/a) + (y/b) = 1 

would work but I couldn't get it in the correct form.  Then I tried to
solve for x and y and got 

  x = 1 + ky 

and 

  y = (-1/k) + (-x/k) 

but now I do not know what to do with this.  So I was wondering if you
knew how to go about solving this? 

Date: 09/16/2003 at 12:28:52
From: Doctor Peterson
Subject: Re: To state the geometric property of an equation

Hi, Mara.

I don't think there is any method you can use to solve this sort of 
problem without a lot of thinking and testing. Let's see how I 
personally would approach it (as well as I can construct it, 
considering that I know the answer already!). Then I'll look at some 
alternative approaches you might take.

We have

  x - ky = 1

and we want to know what property all these lines have in common. 
Probably, since this is an open-ended question and I don't expect it 
to be straightforward, I would start by just "playing" with the 
equation, getting a feel for how it works by trying a few special 
cases. I might take k = 0, 1, and -1 and graph those three lines,

  x = 1
  x - y = 1
  x + y = 1

I would find that they all intersect at (1,0), and my answer would be 
that all the lines seem to contain that point. (In a sense this is a 
more purely "geometric" property than the x-intercept, since it does 
not refer to the coordinates.) Then I would want to prove that this 
is true for ALL k, to make sure I hadn't fooled myself by choosing 
three cases that happened to intersect.

Thinking of this as a point shared by all lines in the family, I 
would prove it by simply substituting x=1, y=0 in the general 
equation:

  x - ky = 1

  1 - k*0 = 1  is true for all k

so (1,0) is indeed on all the lines, not just the three I tried.

Or, I might think of it as a common x-intercept, as you said; then I 
would do what you suggested and put the equation into two-intercept 
form

  x/a + y/b = 1

That's easy; all it takes is to interpret -ky as y divided by -1/k:

  x/1 + y/(-1/k) = 1

So the x-intercept is 1 for all these lines, and the y-intercept is 
-1/k.

So my approach is to experiment (the more adult word for "play"!) and 
make a conjecture (the more adult word for "guess"), and then prove 
that conjecture.

Now, is there any other way you might approach this?

If you were really smart (and I might possibly have done this if I 
were faced with the equation afresh), you could just see that the 
equation looks like the two-intercept form, and gone directly to the 
proof. If you could do that, fine; but you can't depend on such 
insight!

You might instead just go through each form of the equation, starting 
probably with slope-intercept, and see whether any important feature 
(such as the slope or y-intercept) is constant. When that failed, it 
would be hard to move on to the point-slope or two-point form, 
because you would have to choose the point(s), and there is no 
obvious basis for that choice. So you would probably next try the two-
intercept form (which many studuents never see, so you're lucky).

Your approach came close. When you solved for x, you just had to look 
and see that the x-intercept (the constant in that form) is always 1. 
But since that form, the slope-x-intercept form, is little-known, 
it's not surprising that you did not know what to do with it.

But I really think that my approach is the most reasonable hope to 
find the answer quickly.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/