Linear Equations (General and Standard forms: From Wikipedia)

  1. Source: http://en.wikipedia.org/wiki/Linear_equation

    General form:-

    [​IMG]

    It says (under the title General Form) "If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B."

    How is the slope deduced to be -A/B?


    Standard form:-

    [​IMG]

    Aren't this form contradictory to the first form?
    I mean, both are the same, but here C is positive, and on the left side of the equation, while in the General Form, C is positive, and on the right side of the equation.
     
  2. jcsd
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    The slope of a line is the amount that y changes if x increases by 1.
    If x increases by 1, Ax increases by A, so By has to decrease by A.
    More specifically if y changes by -A/B, then By changes by -A.

    The second form is a new equation in which the old A, B and C no longer apply.
    It is more or less related to the first form, but as you already surmised the C in the second form would have to be minus the C of the first form, if A and B are the same.
     
  4. But I thought the change doesn't have to be 1 in x? What if the change is p?
     
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    The slope of a line is defined to be the change in y if x changes by 1.

    If x changes by p, then y will change by p times the slope.
     
  6. From Wikipedia: "The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. "

    That's what I meant.

    "The slope of a line is the amount that y changes if x increases by 1.
    If x increases by 1, Ax increases by A, so By has to decrease by A."

    So, if x increases by p, Ax increase by Ap, so By has to decrease by Ap. (So, y = Ap/B).

    This means -Ap/B should be the slope in general for the equation Ax + By + C = 0. Or is my conclusion wrong?
     
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    As you say, the slope is the change in the y coordinate divided by the corresponding change in the x coordinate.

    Since -Ap/B is the change in y if x changes by p, that means that the slope is -Ap/B divided by p, which is -A/B.
     
  8. That's right.

    Thank you :)
     
  9. I like Serena

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    You're welcome. :smile:
     
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