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Cramer's Rule for 3 x 3's works, pretty much, the same way it does for 2 x 2's -- it's the same pattern.

Let's solve this one:

             2x - y + 6z = 10

 -3x + 4y -5z = 11

        8x - 7y - 9z = 12

First, find the determinant of the coefficient matrix:

(I'm just going to crunch the determinants without
showing the work -- you should check them!)

D = | row 1: 2 , -1 , 6  row 2: -3 , 4 , -5  row 3: 8 , -7 , -9 | = -141

For a 3 x 3, we have 3 more determinants to find: Dx , Dy , and
Dz ...  Then we'll have

 

x = ( Dx / D ) and y = ( Dy / D ) and z = ( Dz / D )