Playing with Equations to Solve ProblemsDate: 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/ |
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