# Cramer's Rule Original Problem

1. Mar 20, 2009

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. Mar 20, 2009

### 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.

3. Mar 20, 2009