Question

Please please please help!! i need this asap and i have no where else to go!

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

x + y + z = 2
2x − 3y + z = −11
− x + 2y − z = 8


A. x = 5, y = 1, z = 6

B. x= 1/3, y = 10/3, z = -5/3

C. x = 3, y = −1, z = 2

D. x = 1, y = 3, z = 2

Answer 1

Option B is correct!

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

Answer 2

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