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Cramer's Rule and Determinants

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Use Cramer's rule to solve the linear system.


    2. Relevant equations
    (only showing one, I think if one is explained I will figure out the rest)
    2x - y = -2
    x + 2y = 14

    What I'm told I'm supposed to do, is to take the constants accompanying the variables and make a matrix out of them (2, -1, 1, 2)
    Then find the determinant. To find the determinant, multiply diagonally down from the top left, and subtract that from the product of multiplying diagonally up from the bottom left. If I'm doing this right, I get 4 - (-1) which would give me 5. I found the determinant, and I have no idea what to do from there.


    3. The attempt at a solution

    What I think I'm supposed to do going off a rough memory, is take a matrix from the equaled values (-2 and 14) and put them on the right of the matrix, then use the Xs on the left. Find the determinant of that, and divide it by 5 (the determinant of the first one.) Of course, I really have no idea.
    So if I do what I think I'm supposed to do (which I'm quite sure is the wrong thing) I get 28 - (-2) and get 30. 30 divided by 5 is 6. I think 6 is y.

    So if y is 6, then 2x - 6 = -2, so 2x = 4. x = 2. So x is 2, x + 2y is 2 + 2(6) which is actually 14.

    So maybe I got it right, but I'd like a bit of confirmation.
     
    Last edited: Mar 14, 2012
  2. jcsd
  3. Mar 14, 2012 #2
    You have the correct idea for the y coefficients. Instead of plugging in y into one of the equations and solving for x, you can just do the same substitution into the first column (leaving the second alone) and calculating its determinant.

    If you're still confused, I can illustrate it with matrices, but everything you did was correct!
     
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