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Math Help - Equation and system in N

  1. #1
    Super Member dhiab's Avatar
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    Equation and system in N

    1 ) sOLVE in this equation : .

    2 ) solve in the system :

    3 ) solve in the system :
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  2. #2
    No one in Particular VonNemo19's Avatar
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    What do you mean by N^2?
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  3. #3
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    Hello, dhiab!

    These are strange problems . . .


    Solve in N^2\!:\quad\begin{array}{ccc}xy & \leq & 2x \\ x+y &=& 4\end{array}

    The line x + y \;=\:4 has intercepts (4,0) and (0,4).
    Code:
              |
            * |
              *
              | *
              |   *
              |     *
          - - + - - - * - -
              |         *
              |

    We have the inequality: . xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0

    This is true when x and (y-2) have opposite signs.

    There are two cases: . \begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}

    The graph looks like this:
    Code:
          ::::::::|
          ::::::::|
          ::::::::|
          ::::::::|
         - - - - 2+ - - - - - -
                  |::::::::::
      ------------+:-:-:-:-:----
                  |:::::::::: 
                  |::::::::::
                  |::::::::::

    Together, we have this graph . . .
    Code:
          ::::*:::|
          ::::::*:|
          ::::::::*
          ::::::::| *
         - - - - 2+ - * - - - -
                  |:::::*:::::::
        ----------+:-:-:-:*:-:-:-
                  |:::::::::*:::
                  |:::::::::::*:
                  |:::::::::::::*

    The final graph looks like this:
    Code:
              *   |
                * |
                  *
                  |
                 2+ - * - - - -
                  |     *
        ----------+-------*------
                  |         *
                  |           *
                  |             *

    How can we describe this graph?

    Maybe: . x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)

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  4. #4
    Super Member dhiab's Avatar
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    Quote Originally Posted by VonNemo19 View Post
    What do you mean by N^2?
    = N ŚN
    Last edited by mr fantastic; June 8th 2009 at 11:49 PM.
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  5. #5
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by dhiab View Post
    = N ŚN
    That is, the set of ordered pairs such that each element in the ordered pair is a natural number.
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  6. #6
    Super Member dhiab's Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, dhiab!

    These are strange problems . . .


    The line x + y \;=\:4 has intercepts (4,0) and (0,4).
    Code:
              |
            * |
              *
              | *
              |   *
              |     *
          - - + - - - * - -
              |         *
              |
    We have the inequality: . xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0

    This is true when x and (y-2) have opposite signs.

    There are two cases: . \begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}

    The graph looks like this:
    Code:
          ::::::::|
          ::::::::|
          ::::::::|
          ::::::::|
         - - - - 2+ - - - - - -
                  |::::::::::
      ------------+:-:-:-:-:----
                  |:::::::::: 
                  |::::::::::
                  |::::::::::
    Together, we have this graph . . .
    Code:
          ::::*:::|
          ::::::*:|
          ::::::::*
          ::::::::| *
         - - - - 2+ - * - - - -
                  |:::::*:::::::
        ----------+:-:-:-:*:-:-:-
                  |:::::::::*:::
                  |:::::::::::*:
                  |:::::::::::::*
    The final graph looks like this:
    Code:
              *   |
                * |
                  *
                  |
                 2+ - * - - - -
                  |     *
        ----------+-------*------
                  |         *
                  |           *
                  |             *
    How can we describe this graph?

    Maybe: . x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)
    Hello : 0n has two cases in question 2 : x=0 and x no zero
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