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Thread: Need help with this equation using reduction methods

  1. #1
    Senior Member
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    Oct 2008
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    393

    Need help with this equation using reduction methods

    For which values of
    k are there no solutions, many solutions, or a unique solution to this
    system?

    x
    y = 1

    3
    x 3y = k

    This is what i did

    1 -1 | 1
    3 -3 | k

    Line 2 - (3 x line 1)

    1 -1 | 1
    0 0 |k-3

    So k = 3 there are inifinte solutions
    And when k doesnt = 3 there are no solutions

    But what about one solution? Is there one?
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  2. #2
    Senior Member
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    Oct 2008
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    Realized i've already asked this exact same question woops, mods delete please. Sorry.
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  3. #3
    Senior Member yeKciM's Avatar
    Joined
    Jul 2010
    Posts
    456
    Quote Originally Posted by adam_leeds View Post
    For which values of
    k are there no solutions, many solutions, or a unique solution to this
    system?

    x
    y = 1

    3
    x 3y = k

    This is what i did

    1 -1 | 1
    3 -3 | k

    Line 2 - (3 x line 1)

    1 -1 | 1
    0 0 |k-3

    So k = 3 there are inifinte solutions
    And when k doesnt = 3 there are no solutions

    But what about one solution? Is there one?
    i hope this is what you wrote :

    $\displaystyle x-y=1$

    $\displaystyle 3x-3y=k$


    there is no unique (one) solution because determinant is not different from zero


    $\displaystyle D=\begin{vmatrix}
    1 &-1 \\
    3 &-3
    \end{vmatrix} = 1\cdot (-3) - 3\cdot (-1) = 0$

    $\displaystyle D_x= \begin{vmatrix}
    1 &-1 \\
    k &-3
    \end{vmatrix} = -3+k = k-3 $

    $\displaystyle D_y= \begin{vmatrix}
    1 &1 \\
    3 &k
    \end{vmatrix} = k-3 $

    so to conclude ... (this is theory that you should know)

    $\displaystyle D\neq 0 \Rightarrow $ there is unique solution

    $\displaystyle \displaystyle x = \frac {D_x}{D}$

    $\displaystyle \displaystyle y = \frac {D_y}{D}$




    $\displaystyle D= 0 \Rightarrow $ and $\displaystyle D_x\neq 0 \vee D_y\neq 0$ there is no solution



    $\displaystyle D= D_x=D_y=0 \Rightarrow $ there is infinite many solutions
    Last edited by yeKciM; Aug 23rd 2010 at 10:51 AM.
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