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Cramer's Rule for 3x3's

Finally, for

 Dz :

You know how this is going to work...

Take D...

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

and delete the column with the z guys...

 | row 1: 2 , -1 , __ row 2: -3 , 4 , __ row 3: 8 , -7 , __ | ... delete the z guys (column 3)

Replace the column with the "= guys"...

Dz = | row 1: 2 , -1 , 10 row 2: -3 , 4 , 11 row 3: 8 , -7 , 12 | = 16
 

So, z = ( Dz / D ) = ( 16 / -141 ) = ( -16 / 141 )


So, our final answer is

( ( 1499 / 141 ) , ( 1492 / 141 ) , ( -16 / 141 ) )

 

Cramer's Rule is great, but crunching a bunch of 3 x 3 determinants takes a long time and there are only about 6 billion places to make mistakes.  On the other hand, doing elimination with messy fractions like these can't be a pretty thing.  The moral here is, if the numbers aren't super clean, solving 3 x 3 is hard!
 

YOUR TURN (and good luck!):

Solve using Cramer's Rule:

2x - 3y = 4

  -x + 4y - z = 11

     6x - 5y + 2z = -3