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Question 135991: I have another question. My professor wants us to solve these equations in four different ways and show our work. I have been able to do the simpler equations but when they have 3 or 4 variables it gets a little bit more complicated.
I need to use the elimination method, substitution method, determinant and coordinates for example if if x=0 y=-5 etc...
Here are two problems that I'm having a hard time with.
#5
x+y+z=6
2x-y+3z=9
-x+2y+z=6
#7
3x+2y+z=6
2x-y+4z=-4
x+y-2z=5
Each one has to be done in four different ways. Thanks again for all the help.
Anne
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! When you say 'coordinate' method, do you mean to solve the system by graphing? I certainly hope not because there is no way to render a three-dimensional coordinate system on this site. However, I'll do one of your systems by the other three methods.
1. 
2. 
3. 
Elimination:
Replace Eq. 2 with the sum of Eq. 2 and two times Eq. 3.
Replace Eq. 3 with the sum of Eq. 1 and Eq. 3.
1.
2. 
3. 
Replace Eq. 3 with the sum of Eq. 2 and -1 times Eq. 3.
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2.
3. 
Divide Eq 3 by 3
1.
2.
3. 
Replace Eq. 1 with the sum of Eq. 1 and -1 times Eq. 3
Replace Eq. 2 with the sum of Eq. 2 and -5 times Eq. 3
1. 
2. 
3.
Divide Eq. 2 by 3
1.
2. 
3.
Replace Eq. 1 with the sum of Eq. 1 and -1 times Eq. 2
1. 
2.
3.
Eliminate the variables with zero coefficients
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2. 
3. 
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Substitution:
1.
2.
3.
Solve Eq. 1 for x:
Substitute the expression for x into the other two equations
2. 
3. 
Distribute and collect like terms:
2.  =>  (New Eq. 2)
3.  =>  (New Eq. 3)
Solve equation 2 for y:
Substitute this expression for y into the new Eq. 3
Distribute, simplify, collect terms
Substitute this value for z into the new Eq. 2
Substitute the values for y and z into the original equation 1:
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Cramer's Rule
Create and evaluate the coefficient determinant
D =
If you don't remember how to evaluate a determinant, here is a little cartoon that shows the process. You have to take your coefficient determinant and make a 5X3 matrix by repeating the 1st and 2nd columns first. Then follow the process shown:
Yours evaluates to: -9, so we say 
Next, replace the first column representing the coefficients on the x terms with the constant term values.
 , and do the determinant evaluation on t
This works out to -9 as well, so you can say 
Cramer's rule says:
 . In this case:  (Just as I most sincerely hope you expected)
As you might suspect, you replace the second column with the coefficients to get  and the third column to get  .
Then  and 
I'll give you the and matrices and you can do the calculations for yourself.
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