SOLUTION: I have tried every solution and I cannot come up with an answer to this problem. I am trying to solve the problem by Crammer's Rule using determinants. The problem is: 3x - 2y+ z

Algebra ->  Matrices-and-determiminant -> SOLUTION: I have tried every solution and I cannot come up with an answer to this problem. I am trying to solve the problem by Crammer's Rule using determinants. The problem is: 3x - 2y+ z       Log On


   

Question 136905: I have tried every solution and I cannot come up with an answer to this problem.
I am trying to solve the problem by Crammer's Rule using determinants. The problem is: 3x - 2y+ z = 6
4x -4y + 3z = 0
5x - 4y + z = -5
I would very much appreciate your help,I have spent hours trying to find solutions to this problem. Thank-you.
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
solve the problem by Cramer's Rule using determinants. The problem is:
3x - 2y+ z = 6
4x -4y + 3z = 0
5x - 4y + z = -5
-------------------------
The determinant of the coefficients is:
[-12-16-30]-[-20-36-8] = -58--64 = 6
-------------------------
Replace the "x-column" with the constant column to get:
6...-2...1
0...-4...3
-5..-4...1
------------
The determinant of that 3 x 3 matrix is:
[-24+0+30)-(20+-72+0) = 6--52= 58
---------------------
Divide the x-determinant by the coefficient determinant to get x = 58/6=29/3
-------------------
To find "y":
1st:Replace the "y-column" in the coefficient matrix by the constant column.
2nd:Find the determinant of this matrix
3rd:Divide that value by 6 to get the value of "y".
---------------------
Follow the same pattern for "z".
---------------------
Final answer:
x = 29/6
y = 91/6
z = 44/6
==============
Cheers,
Stan H.


Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1: Create the coefficient determinant and the answer column matrix

D=%28matrix%283%2C3%2C3%2C-2%2C1%2C4%2C-4%2C3%2C5%2C-4%2C1%29%29
AC=%28matrix%283%2C1%2C6%2C0%2C-5%29%29

Step 2: Evaluate the determinant.





Step 3: Now, take the coefficient determinant and replace the x-column (the first column) with the answer column, like this:

D%5Bx%5D=%28matrix%283%2C3%2C6%2C-2%2C1%2C0%2C-4%2C3%2C-5%2C-4%2C1%29%29

Step 4: Evaluate the D%5Bx%5D determinant as shown in step 2.

Cramer's rule states that x=D%5Bx%5D%2FD

Step 5: Repeat step 3 and 4, except replace the y-column (the second column) with the answer column matrix and apply Cramer's rule: y=D%5By%5D%2FD

Step 6: Do it one more time replacing the z-column (the third column) with the answer column. z=D%5Bz%5D%2FD

Hint: Be extremely meticulous about the signs on the numbers when evaluating the determinants. If you have access to Microsoft Excel, you can put the coefficients into a square set of cells, one coefficient per cell. Let's say you put them in cells A1 through C3, then in an empty cell, type in =MDETERM(A1:C3). When you hit enter, the value of the determinant will appear.