Problem to solve
Solve \[2x-\frac{y}{3}+2z=-1\], \[x+y+\frac{z}{2}=-1\], \[x-2y+z=2\]
Solution
STEP 1
Solve the following system:
{2 x - y/3 + 2 z = -1
x + y + z/2 = -1
x - 2 y + z = 2
STEP 2
Express the system in matrix form:
(2 | -1/3 | 2
1 | 1 | 1/2
1 | -2 | 1)(x
y
z) = (-1
-1
2)
STEP 3
Write the system in augmented matrix form and use Gaussian elimination:
(2 | -1/3 | 2 | -1
1 | 1 | 1/2 | -1
1 | -2 | 1 | 2)
STEP 4
Subtract 1/2 × (row 1) from row 2:
(2 | -1/3 | 2 | -1
0 | 7/6 | -1/2 | -1/2
1 | -2 | 1 | 2)
STEP 5
Multiply row 1 by 3:
(6 | -1 | 6 | -3
0 | 7/6 | -1/2 | -1/2
1 | -2 | 1 | 2)
STEP 6
Multiply row 2 by 6:
(6 | -1 | 6 | -3
0 | 7 | -3 | -3
1 | -2 | 1 | 2)
STEP 7
Subtract 1/6 × (row 1) from row 3:
(6 | -1 | 6 | -3
0 | 7 | -3 | -3
0 | -11/6 | 0 | 5/2)
STEP 8
Multiply row 3 by 6:
(6 | -1 | 6 | -3
0 | 7 | -3 | -3
0 | -11 | 0 | 15)
STEP 9
Swap row 2 with row 3:
(6 | -1 | 6 | -3
0 | -11 | 0 | 15
0 | 7 | -3 | -3)
STEP 10
Add 7/11 × (row 2) to row 3:
(6 | -1 | 6 | -3
0 | -11 | 0 | 15
0 | 0 | -3 | 72/11)
STEP 11
Multiply row 3 by 11/3:
(6 | -1 | 6 | -3
0 | -11 | 0 | 15
0 | 0 | -11 | 24)
STEP 12
Divide row 3 by -11:
(6 | -1 | 6 | -3
0 | -11 | 0 | 15
0 | 0 | 1 | -24/11)
STEP 13
Divide row 2 by -11:
(6 | -1 | 6 | -3
0 | 1 | 0 | -15/11
0 | 0 | 1 | -24/11)
STEP 14
Add row 2 to row 1:
(6 | 0 | 6 | -48/11
0 | 1 | 0 | -15/11
0 | 0 | 1 | -24/11)
STEP 15
Subtract 6 × (row 3) from row 1:
(6 | 0 | 0 | 96/11
0 | 1 | 0 | -15/11
0 | 0 | 1 | -24/11)
STEP 16
Divide row 1 by 6:
(1 | 0 | 0 | 16/11
0 | 1 | 0 | -15/11
0 | 0 | 1 | -24/11)
STEP 17
Collect results:
Answer: | 
 | {x = 16/11
y = -15/11
z = -24/11