Problem to solve
Solve \[x-y+3z=2\], \[2x-3y=-4\], \[x+y+z=-2\]
Solution
STEP 1
Solve the following system:
{x - y + 3 z = 2
-3 y + 2 x = -4
x + y + z = -2
STEP 2
Express the system in matrix form:
(1 | -1 | 3
2 | -3 | 0
1 | 1 | 1)(x
y
z) = (2
-4
-2)
STEP 3
Write the system in augmented matrix form and use Gaussian elimination:
(1 | -1 | 3 | 2
2 | -3 | 0 | -4
1 | 1 | 1 | -2)
STEP 4
Swap row 1 with row 2:
(2 | -3 | 0 | -4
1 | -1 | 3 | 2
1 | 1 | 1 | -2)
STEP 5
Subtract 1/2 × (row 1) from row 2:
(2 | -3 | 0 | -4
0 | 1/2 | 3 | 4
1 | 1 | 1 | -2)
STEP 6
Multiply row 2 by 2:
(2 | -3 | 0 | -4
0 | 1 | 6 | 8
1 | 1 | 1 | -2)
STEP 7
Subtract 1/2 × (row 1) from row 3:
(2 | -3 | 0 | -4
0 | 1 | 6 | 8
0 | 5/2 | 1 | 0)
STEP 8
Multiply row 3 by 2:
(2 | -3 | 0 | -4
0 | 1 | 6 | 8
0 | 5 | 2 | 0)
STEP 9
Swap row 2 with row 3:
(2 | -3 | 0 | -4
0 | 5 | 2 | 0
0 | 1 | 6 | 8)
STEP 10
Subtract 1/5 × (row 2) from row 3:
(2 | -3 | 0 | -4
0 | 5 | 2 | 0
0 | 0 | 28/5 | 8)
STEP 11
Multiply row 3 by 5/4:
(2 | -3 | 0 | -4
0 | 5 | 2 | 0
0 | 0 | 7 | 10)
STEP 12
Divide row 3 by 7:
(2 | -3 | 0 | -4
0 | 5 | 2 | 0
0 | 0 | 1 | 10/7)
STEP 13
Subtract 2 × (row 3) from row 2:
(2 | -3 | 0 | -4
0 | 5 | 0 | -20/7
0 | 0 | 1 | 10/7)
STEP 14
Divide row 2 by 5:
(2 | -3 | 0 | -4
0 | 1 | 0 | -4/7
0 | 0 | 1 | 10/7)
STEP 15
Add 3 × (row 2) to row 1:
(2 | 0 | 0 | -40/7
0 | 1 | 0 | -4/7
0 | 0 | 1 | 10/7)
STEP 16
Divide row 1 by 2:
(1 | 0 | 0 | -20/7
0 | 1 | 0 | -4/7
0 | 0 | 1 | 10/7)
STEP 17
Collect results:
Answer: | 
 | {x = -20/7
y = -4/7
z = 10/7