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Use Gauss-Jordan elimination to solve the following system of equations.


x + y + z = 2

2x − 3y + z = −11

− x + 2y − z = 8


Ax = 5, y = 1, z = 6

Bx= 1/3, y = 10/3, z = -5/3

Cx = 3, y = −1, z = 2

Dx = 1, y = 3, z = 2

Verified Answer :

nobillionaireNobley

1   1   1        2
2  -3   1      -11            -2R1 + R2 → R2
-1  2  -1        8                R1 + R3 → R3

1    1    1      2
0    -5  -1   -15             R2 ⇔ R3
0    3    0     10

1    1    1      2
0    3    0     10             -R3
0   -5   -1    -15

1    1    1      2
0    3    0     10             1/3 R2
0    5    1     15

1    1    1      2                   -R2 + R1
0    1    0     10/3             -5R2 + R3
0    5    1     15

1    0    1      -4/3              
0    1    0     10/3             -R3 + R1
0    0    1      -5/3

1    0    0       1/3             
0    1    0     10/3           
0    0    1      -5/3

Therefore, x = 1/3, y = 10/3, z = -5/3

Answer:

Option B is correct!

Step-by-step explanation:


solving using Gauss-Jordan elimination


      x y z b

1 1 1 1 2

2 2 -3 1 -11

3 -1 2 -1 8

Finding  the pivot in the 1st column in the 1st row


       x y z b

1 1 1 1 2

2 2 -3 1 -11

3 -1 2 -1 8

Multiplying  the 1st row by 2


       x y z b

1 2 2 2 4

2 2 -3 1 -11

3 -1 2 -1 8

Subtract the 1st row from the 2nd row and restore it


       x y z b

1 1 1 1 2

2 0 -5 -1 -15

3 -1 2 -1 8

Multiplying  the 1st row by -1


        x y z b

1 -1 -1 -1 -2

2 0 -5 -1 -15

3 -1 2 -1 8

Subtract the 1st row from the 3rd row and restore it


       x y z b

1 1 1 1 2

2 0 -5 -1 -15

3 0 3 0 10

Make the pivot in the 2nd column by dividing the 2nd row by -5


      x y z b

1 1 1 1 2

2 0 1 1/5 3

3 0 3 0 10

Subtracting  the 2nd row from the 1st row


       x y z b

1 1 0 4/5 -1

2 0 1 1/5 3

3 0 3 0 10

Multiplying the 2nd row by 3


      x y z b

1 1 0 4/5 -1

2 0 3 3/5 9

3 0 3 0 10

Subtracting  the 2nd row from the 3rd row


       x  y     z       b

1 1 0 4/5 -1

2 0 1 1/5 3

3 0 0 -3/5 1

Making  the pivot in the 3rd column by dividing the 3rd row by -3/5


      x y z b

1 1 0 4/5 -1

2 0 1 1/5 3

3 0 0 1 -5/3

Multiplying the 3rd row by 4/5


      x y z b

1 1 0 4/5 -1

2 0 1 1/5 3

3 0 0 4/5 -4/3

Subtracting the 3rd row from the 1st row and restore it


      x      y z b

1 1 0 0 1/3

2 0 1 1/5 3

3 0 0 1 -5/3

Multiply the 3rd row by 1/5


      x y z b

1 1 0 0 1/3

2 0 1 1/5 3

3 0 0 1/5 -1/3

Subtract the 3rd row from the 2nd row and restore it


      x y z b

1 1 0 0 1/3

2 0 1 0 10/3

3 0 0 1 -5/3

Here


x = 1/3

y = 10/3

z=-5/3

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