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How to express solution to system of equations

  1. Jan 25, 2009 #1
    Sorry if this is too basic for this forum, but it IS from a 2nd-year linear algebra course. I'm probably just being stupid, and missing something obvious, but here goes:

    The problem statement, all variables and given/known data

    Solve the following system of equations:

    [tex]2x_{1} - 2x_{2} - 3x_{3} = -2[/tex]
    [tex]3x_{1} - 3x_{2} - 2x_{3} + 5x_{4} = 7[/tex]
    [tex]x_{1} - x_{2} - 2x_{3} - x_{4} = -3[/tex]

    The attempt at a solution

    So, clearly there won't be a unique solution. I do the work and get this matrix:
    [tex]
    \[ \left( \begin{array}{cccc|c}
    1 & -1 & 0 & 0 & 5 \\
    0 & 0 & 1 & 0 & 4 \\
    0 & 0 & 0 & 1 & 0 \end{array} \right)\]
    [/tex]

    However, the answer in the back of the book is
    {r(1,1,0,0) + s(-3,0,-2,1) + (5,0,4,0) | r,s [tex]\in \mathbb{R}[/tex]}

    My question is how to get this from the solution matrix. Regardless of whether the solution is right or wrong, I don't know where this is coming from. I can conceivably see that (5,0,4,0) can be obtained by choosing x_2 = 0, but that's about it.

    Thanks!
     
    Last edited: Jan 25, 2009
  2. jcsd
  3. Jan 26, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Are you sure you don't get something else, like, say,
    [tex]
    \[ \left( \begin{array}{cccc|c}
    2 & -2 & 0 & 3 & 5 \\
    0 & 0 & 1 & 2 & 4 \\
    0 & 0 & 0 & 0 & 0 \end{array} \right)\]
    [/tex]
     
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