SOLUTION: solve the following systems of linear equation in three variables by elimination or substitution. 1. x+y+z=6 2x-y+3z=9 -x+2y+2z=9

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Question 662393: solve the following systems of linear equation in three variables by elimination or substitution.
1. x+y+z=6
2x-y+3z=9
-x+2y+2z=9
Found 2 solutions by ReadingBoosters, Edwin McCravy:
Answer by ReadingBoosters(3246) About Me  (Show Source):
You can put this solution on YOUR website!
Starting with
x+y+z=6
2x-y+3z=9
-x+2y+2z=9
Using elimination
multiply the first equation by -1
-x-y-z=-6
Add all the equations together
2x-y+3z=9
-x-y-z=-6
-x+2y+2z=9
such that
2x+-x+-x+-y+-y+2y+3z+-z+2z=9+-6+9
combine like terms
4z = 12
z=3
using z, add the original first two equations
x+2x+y+-y+z+3z=6+9
combine like terms and substitute z
3x+4z=15
3x+4(3)=15
3x=15-12
x=1
using x and z, add the original first and third equations
x+-x+y+2y+z+2z=6+9
3y+3z=15
3y+3(3)=15
3y = 15-9
y=2
so, x=1, y=2, z=3
Proof
3+1+2=6
2(1)-2+3(3)=9
-1+2(2)+2(3)=9

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
1)      x +  y +  z = 6
2)     2x -  y + 3z = 9
3)     -x + 2y + 2z = 9

Pick a letter to eliminate and a pair of equations that contain
it and eliminate it from them:

I choose to eliminate x from 1) and 3):

1)      x +  y +  z =  6
3)     -x + 2y + 2z =  9  Just add them term by term
------------------------
            3y + 3z = 15  Divide thru by 3
4)           y +  z =  5

Pick one of the equations that you just used along
with the equation you haven't used and eliminate the
SAME letter you just eliminated. 

I pick 1) and 2) to eliminate x from:

1)      x +  y +  z = 6
2)     2x -  y + 3z = 9 
----------------------- Multiply 10 thru by -2

      -2x - 2y - 2z = -12
2)     2x -  y + 3z =   9 
-------------------------
5)         -3y +  z =  -3

Solve the resulting 2x2 system of equations:

Now we take 4) and 5)

4)           y +  z =  5
5)         -3y +  z = -3

Eliminate z by multiplying 4) by -1

            -y -  z = -5
5)         -3y +  z = -3
------------------------
6)         -4y      = -8
                  y =  2

Substitute in 4)

4)            y + z = 5
              2 + z = 5
                  z = 3

 Substitute y = 2 and z = 3 in 1)

1)        x + y + z = 6
          x + 2 + 3 = 6
              x + 5 = 6
                  x = 1

Edwin