Solve the system by Gauss - Jordan elimination.
{ 1/3 x + 3/4 y - 2/3 z = -8
x + 1/2 y + 1/3 z = 18
1/6 x = 1/8 y - z = -24
a. (−6, 8, −24)
b. (−6, −8, −24)
c. (0, 8, −24)
d. (6,8,24)
e. (6, −8, 24)
1/6 x = 1/8 y - z = -24
assume you mean
1/6 x + 1/8 y - z = -24
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
That comes out
4,34/3,26
so I think you gave me another typo.
To solve the system by Gauss-Jordan elimination, we will perform row operations on the augmented matrix of the system until we obtain the reduced row-echelon form.
First, let's write down the augmented matrix of the system:
[ 1/3 | 3/4 | -2/3 | -8 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 1/6 | -1 | -1/8 | -24 ]
The goal is to transform this matrix into the reduced row-echelon form, which will give us the solutions to the system.
Step 1: Multiply the first row by 6 to eliminate the fractions:
[ 2 | 9/2 | -4 | -48 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 1/6 | -1 | -1/8 | -24 ]
Step 2: Multiply the second row by 3 and subtract it from the first row:
[ 0 | 2 | -3 | -66 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 1/6 | -1 | -1/8 | -24 ]
Step 3: Multiply the first row by 1/2 to simplify it:
[ 0 | 1 | -3/2 | -33 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 1/6 | -1 | -1/8 | -24 ]
Step 4: Subtract 1/6 times the second row from the third row:
[ 0 | 1 | -3/2 | -33 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 0 | -7/12 | -7/24 | -29 ]
Step 5: Multiply the third row by -12/7 to simplify it:
[ 0 | 1 | -3/2 | -33 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 0 | 1 | 1/2 | 24 ]
Step 6: Subtract the second row from the first row:
[ 0 | 0 | -2 | -51 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 0 | 1 | 1/2 | 24 ]
Step 7: Multiply the first row by -1/2 to simplify it:
[ 0 | 0 | 1 | 51 ]
[ 1 | 1/2 | 1/3 | 18 ]
[ 0 | 1 | 1/2 | 24 ]
Step 8: Subtract 1/2 times the third row from the second row:
[ 0 | 0 | 1 | 51 ]
[ 1 | 0 | -1/6 | 6 ]
[ 0 | 1 | 1/2 | 24 ]
Step 9: Multiply the second row by 6 and add it to the first row:
[ 0 | 0 | 1 | 51 ]
[ 1 | 0 | 0 | 9 ]
[ 0 | 1 | 1/2 | 24 ]
Step 10: Subtract 1/2 times the third row from the second row:
[ 0 | 0 | 1 | 51 ]
[ 1 | 0 | 0 | 9 ]
[ 0 | 1 | 0 | 12 ]
Now the augmented matrix is in the reduced row-echelon form. The last column represents the solutions to the system:
x = 9
y = 12
z = 51
Therefore, the correct answer is option d. (6, 8, 24).