SOLUTION: 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

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.
1.
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
1.
2.
3.

========================================================
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:

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