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Homework Help: System of linear equations with alpha variable

  1. May 27, 2010 #1
    1. The problem statement, all variables and given/known data

    For which value of alpha does the following system of linear equations have a solution?

    x + y = 2
    x +αy = -1
    αx + y = 1


    3. The attempt at a solution

    I put it into a matrix that looks like this:

    1 1 | 2
    1 α | -1
    α 1 | 1

    Then I subtracted row1 from row2 and it made the matrix look like:

    1 1 | 2
    0 (α-1) | -1
    α 1 | 1

    Then I took the 2nd row as α -1 = -1 and solved α to be zero. So, does this mean that the solution for this is all numbers α such that α is not equal to zero?
     
  2. jcsd
  3. May 27, 2010 #2

    gabbagabbahey

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    First, when you subtract the first row from the second you get

    [tex]\left[\begin{array}{cc|c} 1 & 1 & 2 \\ 0 & \alpha-1 & -3 \\ \alpha & 1 & 1 \end{array}\right] [/tex]


    Second, your second row tell you [itex](\alpha-1)y=-3[/itex], not [itex](\alpha-1)=-3[/itex]

    No.
     
  4. May 27, 2010 #3
    Cool, thanks! So what's the next step? Not sure where to go from here...
     
  5. May 27, 2010 #4
    Can I get a hint?
     
  6. May 27, 2010 #5

    gabbagabbahey

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    I think the easiest method is to simply say that if [itex](\alpha-1)y=-3[/itex], then [itex]y=\frac{3}{1-\alpha}[/itex] and substitute that into any of your 3 equations, solve for [itex]x[/itex] and sub those into the remaining equations to get an equation for [itex]\alpha[/itex].
     
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