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Math Help - Need help with this equation using reduction methods

  1. #1
    Senior Member
    Joined
    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
    Joined
    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 :

     x-y=1

     3x-3y=k


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


    D=\begin{vmatrix}<br />
1 &-1 \\ <br />
3 &-3 <br />
\end{vmatrix} = 1\cdot (-3) - 3\cdot (-1) = 0

     D_x= \begin{vmatrix}<br />
1 &-1 \\ <br />
k &-3 <br />
\end{vmatrix} = -3+k = k-3

     D_y= \begin{vmatrix}<br />
1 &1 \\ <br />
3 &k <br />
\end{vmatrix} = k-3

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

     D\neq 0 \Rightarrow there is unique solution

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

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




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



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