Question 858091
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Hi, there--

Your Problem:
Solve the following system of linear equations using Cramer's Rule.
{{{x+y-2z=3}}}
{{{3x-y+z=5}}}
{{{3x+3y-6z=9}}}

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.
*[illustration Coefficient_matrix_and_answer_column]


Step 2: Now we find find the determinants of four matrices, {{{D}}}, {{{D[x]}}}, {{{D[y]}}}, and {{{D[z]}}}.

Matrix D is the coefficient matrix.
{{{
   matrix(3, 3, 1, 1, 2, 3, -1, 1, 3, 3, -6) 
}}}

The determinant, |D| = 48
 
Matrix Dx is the coefficient matrix with column one replaced with the answer column.
{{{
   matrix(3, 3, 3, 1, 2, 5, -1, 1, 9, 3, -6) 
}}}
The determinant, |Dx| = 96

Matrix Dy is the coefficient matrix with column two replaced with the answer column.
{{{
   matrix(3, 3, 1, 3, 2, 3, 5, 1, 3, 9, -6)
}}}
The determinant, |Dy| = 48

Matrix Dz is the coefficient matrix with column three replaced with the answer column.
{{{
   matrix(3, 3, 3, 1, 2, 5, -1, 1, 9, 3, -6)
}}}
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
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