SOLUTION: I need help solving this problem using the Cramer's rule to find the value of "z" that satisfies the system of linear equations. Here is the link to my question: http://s699.ph

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Question 255688: I need help solving this problem using the Cramer's rule to find the value of "z" that satisfies the system of linear equations.
Here is the link to my question:
http://s699.photobucket.com/albums/vv356/flowerpatch_01/?action=view¤t=Picture4-10.png
So far I have D= 4,-5, 1
2, 5, -2
-2, 3, -4
Thanks so much in advance for your help!
Found 2 solutions by jim_thompson5910, Edwin McCravy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables



system%284%2Ax%2B0%2Ay%2B-5%2Az=1%2C2%2Ax%2B5%2Ay%2B-2%2Az=-5%2C-2%2Ax%2B3%2Ay%2B-4%2Az=5%29



First let A=%28matrix%283%2C3%2C4%2C0%2C-5%2C2%2C5%2C-2%2C-2%2C3%2C-4%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are 1, -5, and 5 and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=-136. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



---------------------------------------------------------



Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).






Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=186. Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28186%29%2F%28-136%29=-93%2F68



So the first solution is x=-93%2F68




---------------------------------------------------------


We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C4%2C0%2C-5%2C2%2C5%2C-2%2C-2%2C3%2C-4%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).






Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=132.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28132%29%2F%28-136%29=-33%2F34



So the second solution is y=-33%2F34




---------------------------------------------------------





Let's reset again by letting A=%28matrix%283%2C3%2C4%2C0%2C-5%2C2%2C5%2C-2%2C-2%2C3%2C-4%29%29 which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bz%5D






Now compute the determinant of A%5Bz%5D to get abs%28A%5Bz%5D%29=176.



To find the third solution, divide the determinant of A%5Bz%5D by the determinant of A to get: z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28176%29%2F%28-136%29=-22%2F17



So the third solution is z=-22%2F17




====================================================================================

Final Answer:




So the three solutions are x=-93%2F68, y=-33%2F34, and z=-22%2F17 giving the ordered triple (-93/68, -33/34, -22/17)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
I need help solving this problem using the Cramer's rule to find the value of "z" that satisfies the system of linear equations.
Here is the link to my question:
http://s699.photobucket.com/albums/vv356/flowerpatch_01/?action=view¤t=Picture4-10.png
So far I have D= 4,-5, 1
2, 5, -2
-2, 3, -4
Thanks so much in advance for your help!

That photobucket site does not come up on my computer, so
I just made up the numbers on the right, colored red.
You can change them to the right numbers:

I'll assume your system of equations was this:

system%284x-5y%2Bz=red%286%29%2C2x%2B5y-2z=red%283%29%2C-2x%2B3y-4z=red%287%29%29

D=abs%28matrix%283%2C3%2C%0D%0A4%2C-5%2C+1%2C%0D%0A2%2C+5%2C+-2%2C%0D%0A-2%2C+3%2C+-4%29%29 

x is the FIRST unknown and so D%5Bx%5D is just like D
except that the FIRST column is replaced by the three red numbers,
which I made up: 



y is the SECOND unknown and so D%5By%5D is just like D
except that the SECOND column is replaced by the three red numbers,
which I made up: 




z is the THIRD unknown and so D%5Bz%5D is just like D
except that the THIRD column is replaced by the three red numbers,
which I made up: 



Do you know how to evaluate a 3x3 determinant?
If not, post again asking how to.  I will assume
you already know how.

D=-100
D%5Bx%5D=-100
D%5By%5D=100
D%5Bz%5D=300

Then the solutions are:

x=D%5Bx%5D%2FD=%28-100%29%2F100+=+-1
y=D%5By%5D%2FD=%28100%29%2F100+=+1
z=D%5Bz%5D%2FD=%28300%29%2F100+=+3

Now substitute your numbers for the red numbers
and evaluate the determinants and you'll have your 
solution.

Edwin