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Cramer's Rule Original Problem

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Let the curve

    A + By + Cx + Dy^2 + Exy + x^2 = 0

    be given. It passes through the points (x_1, y_1),...,(x_5, y_5). Determine the A, B, C, D, and E.

    2. Relevant equations



    3. The attempt at a solution

    To create the system, I plug in each x_n and y_n into the given curve equation...

    [1, y_1, x_1, (y_1)^2, (x_1)(y_1), (x_1)^2]
    [1, y_2, x_2, (y_2)^2, (x_2)(y_2), (x_2)^2]
    [1, y_3, x_3, (y_3)^2, (x_3)(y_3), (x_3)^2]
    [1, y_4, x_4, (y_4)^2, (x_4)(y_4), (x_4)^2]
    [1, y_5, x_5, (y_5)^2, (x_5)(y_5), (x_5)^2]

    But, what do I do from here? Replace a row with vector b (all 0's in this case) and solve x = determinant of new matrix/determinant of original matrix. Then repeat for each row? Does the fact that there are x^2 and y^2 in the matrix matter at all or not?
     
  2. jcsd
  3. Mar 20, 2009 #2

    Mark44

    Staff: Mentor

    It doesn't matter that the variables x, y, x^2, y^2, and xy aren't linear. You're solving for the constants A, B, C, D, and E, and your equations are linear in these constants.

    When you substitute the known x and y values into the equation, your first equation will be:
    A + y_1*B + x_1*C + (y_1)^2 * D + (x_1)(y_1)*E = -(x_1)^2

    All five equations will look like this but will involve the other four pairs of x and y values.

    Your matrix will be a 5 row by 6 column augmented matrix that you can solve either by row reduction or by the use of Cramer's rule.
     
  4. Mar 20, 2009 #3
    Thank you!
     
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