The slope of the ratio of two lines

Discussion in 'Geometry and Trigonometry' started by StockholmEd, Dec 22, 2022.

  1. StockholmEd

    StockholmEd

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    Hi, folks!

    I need to check that given two lines both of whose slope is negative, then dividing one line by the other yields a third line whose slope is positive. I tried tinkering around with the algebra, but I can't seem to prove it.

    Basically given two lines, y1 = m1x + c1, and y2 = m2x + c2, and given that m1 and m2 are both < 0, I thought it would be easy to show that if I divide the first line by the second, then the result would itself be a line whose slope is > 0.

    But I can't seem to work it out.

    I also tried selecting two points on both lines, via m = (y2 - y1)/(x2 - x1) and simply dividing the coordinates and then trying to create the line formula, but I can't separate the slope of the resulting equations into a form that depends only on x (it keeps getting mixed-up with y and the the c1, c2 constants).

    Am I trying something that makes no sense?

    Thanks,

    Ed
     
    StockholmEd, Dec 22, 2022
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  2. StockholmEd

    Phrzby Phil

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    What do you mean by "dividing one line by the other" ?

    If you mean forming the rational function (m1x + c1)/(m2x + c2), then in general you will not get a linear function, and therefore not a straight line, and therefore not a slope.

    Example: graph (-x+5) / (-2x+7)

    (BTW - why do you use c1 and c2 when conventional terminology is b1 and b2 ?)
     
    Phrzby Phil, Dec 23, 2022
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  3. StockholmEd

    StockholmEd

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    Oh, thank you! I hadn't thought of just plugging it into some graphing software.

    Very interesting.

    (And I use c1/c2 because that's what was in maths books 40 years ago; had no idea b1/b2 was the convention now.)
     
    StockholmEd, Dec 23, 2022
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  4. StockholmEd

    Phrzby Phil

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    So did you mean (m1x + c1)/(m2x + c2) ?

    Just curious - what is the actual problem, or was this just a curiosity - also a legitimate pursuit.

    Also, only books I've ever seen say y = mx+b, where m = slope and b = y-intercept.
     
    Phrzby Phil, Dec 23, 2022
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  5. StockholmEd

    StockholmEd

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    Yes indeed: I did mean (m1x + c1)/(m2x + c2).

    I'm a programmer and trying to calculate the probability of a function's being updated based on its size.

    "Also, only books I've ever seen say y = mx+b, where m = slope and b = y-intercept."- Ah. I'm sure I'm misremembering then. It's been an age since I last read this stuff.
     
    StockholmEd, Dec 23, 2022
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