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Math Help - For what value(s) of k does the system have....?

  1. #1
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    For what value(s) of k does the system have....?

    i) For what value(s) of k does the system have;
    no solutions, a unique solution, infinitely many solutions?

    x + 2y - z = -3
    0x + y - k-3 = -5
    0x + 0y + k^2 -2k = 5k + 11


    ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

    I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k.
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  2. #2
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    Re: For what value(s) of k does the system have....?

    Quote Originally Posted by figleaf7 View Post
    i) For what value(s) of k does the system have;
    no solutions, a unique solution, infinitely many solutions?

    x + 2y - z = -3
    0x + y - k-3 = -5
    0x + 0y + k^2 -2k = 5k + 11
    No wonder you are confused- either you copied this incorrectly or it was given badly. I suspect it was supposed to be
    x+ 2y- z= -3
    y- (k-3)z= -5
    (k^2- 2k)z= 5k+ 11

    If k^2- 2k\ne 0 then z= \frac{5k+ 11}{k^2- 2k}
    Now what happens if k^2- 2k= 0 but 5k+ 11\ne 0? What if both are 0?


    ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

    I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k. [/QUOTE]
    Thanks from figleaf7
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  3. #3
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    Re: For what value(s) of k does the system have....?

    Quote Originally Posted by figleaf7 View Post
    i) For what value(s) of k does the system have;
    no solutions, a unique solution, infinitely many solutions?

    x + 2y - z = -3
    0x + y - k-3 = -5
    0x + 0y + k^2 -2k = 5k + 11


    ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

    I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k.
    In matrix form this system is

    $\displaystyle \begin{align*} \left[ \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix}\right] \left[ \begin{matrix} x \\ y\\ z \end{matrix} \right] &= \left[ \begin{matrix} -3 \\ -5 \\ \phantom{-} 11 \end{matrix} \right] \end{align*}$

    This matrix equation has a unique solution where $\displaystyle \begin{align*} \left| \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix} \right| \neq 0 \end{align*}$.

    It will either have no solution or infinite solutions where the determinant IS 0, so once you know these k values you will then need to investigate the system with those values plugged in.
    Thanks from figleaf7
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    Re: For what value(s) of k does the system have....?

    Yes, you're right, I copied it poorly. Sorry, was on night shift.
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