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I Simplyfiying y^2 = x^2

  1. Mar 6, 2017 #1
    Say we have that ##y^2 = x^2##. Then if we take the square root of both sides, it would seem that we have ##|y| = |x|##. Why does this imply that that ##y=x## or ##y=-x##, rather than implying that ##y=|x|## or ##y=- |x|##?
  2. jcsd
  3. Mar 6, 2017 #2

    Andrew Mason

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    The two answers are equivalent.

    We agree that a = |x| means that
    a = x or
    a = -x.

    If you let a = |y| then
    (1) |y| = x or
    (2) |y| = -x

    But (1) x = |y| means x = y or x = -y and (2) -x = |y| means -x = y or -x = -y (i.e. x = y). So, y = x or y = -x.

  4. Mar 7, 2017 #3


    Staff: Mentor

    Why not just do this?
    ##y^2 = x^2 \Leftrightarrow y^2 - x^2 = 0 \Leftrightarrow (y - x)(y + x) = 0 \Leftrightarrow y = x \text{ or } y = -x##
  5. Mar 7, 2017 #4


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    2016 Award

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    Yet another approach:
    If |x|=|y|, there are four cases:
    x and y positive: then x=y
    x positive, y negative: x=-y
    x negative, y positive: x=-y
    x and y negative: x=y
    Combined, x=y or x=-y. In other words, the two variables are identical up to a possible difference in their sign.

    I neglected the option x=y=0 here, but that fits to the answer as well.
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