SOLUTION: I need a example of how to use the Cramer Rule step by step.

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Question 114878: I need a example of how to use the Cramer Rule step by step.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Hard to tell where to start with this one. I'll just assume that you know how to evaluate a determinant -- if that isn't true, write back and we'll take a step back.

Let's say you have a system of three equations:

a%5B1%5Dx%2Bb%5B1%5Dy%2Bc%5B1%5Dz=d%5B1%5D
a%5B2%5Dx%2Bb%5B2%5Dy%2Bc%5B2%5Dz=d%5B2%5D
a%5B3%5Dx%2Bb%5B3%5Dy%2Bc%5B3%5Dz=d%5B3%5D

First, create the coefficient Determinant,
, and the constant matrix
%28matrix%283%2C1%2Cgreen%28d%5B1%5D%29%2Cgreen%28d%5B2%5D%29%2Cgreen%28d%5B3%5D%29%29%29

Now, replace the first column in the coefficient Determinant with the values in the constant matrix to get the D%5Bx%5D determinant,


Do this two more times, replacing the second and third columns with the constant matrix values to get the D%5By%5D and D%5Bz%5D determinants.




Now, evaluate all four determinants, D, D%5Bred%28x%29%5D, D%5Bgreen%28y%29%5D, and D%5Bblue%28z%29%5D (I've already assumed that you know how to do this, so I'm continuing with Cramer's rule.)

Cramer's Rule says:
x=D%5Bred%28x%29%5D%2FD
y=D%5Bgreen%28y%29%5D%2FD
z=D%5Bblue%28z%29%5D%2FD

All you have to do is the arithmetic.

Hope this helps.

John


P.S. I forgot to mention one very important detail, if the coefficient determinant is zero D=0, then Cramer's Rule doesn't work because you can't divide by zero. The problem is you may still have to figure out if you have an inconsistent system (parallel or possibly skew lines with no solution) or a consistent and undetermined system (coincident lines) with an infinite number of solutions. Cramer's rule cannot answer this question, and you will have to find some other solution method.