Hi, there--

Your Problem:
Solve the following system of linear equations using Cramer's Rule.




Solution:
Cramer's Rule is a solution method that uses matrices and their determinants.

Step 1: Write the coefficient matrix and the answer column for your system.



Step 2: Now we find find the determinants of four matrices, , , , and .

Matrix D is the coefficient matrix.


The determinant, |D| = 48
 
Matrix Dx is the coefficient matrix with column one replaced with the answer column.

The determinant, |Dx| = 96

Matrix Dy is the coefficient matrix with column two replaced with the answer column.

The determinant, |Dy| = 48

Matrix Dz is the coefficient matrix with column three replaced with the answer column.

The determinant, |Dz| = 0


Step 3. Now that we have these values, we can find x, y, and z.

x = Dx / D = 96/48 = 2
y = Dy / D = 48/48 = 1
z = Dz / D = 0/48 = 0

Check your answer by substituting 2 for x, 1 for y, and 0 for z in the original equations. 

2x + y - z = 3
(2) + (1) - 2(0) = 3
2 + 1 = 3
3 = 3
Check!

3x - y + z = 5
3(2) - (1) + (0) = 5
6 - 1 = 5
5 = 5
Check!

3x + 3y - 6z = 9
3(2) + 3(1) - 6(0) = 9
6 + 3 = 9
9 = 9
Check!

Hope this helps. If you have questions about how to find the determinant of a matrix, here is a nice explanation at PurpleMath:
http://www.purplemath.com/modules/determs2.htm

Mrs. Figgy
math.in.the.vortex@gmail.com