Problem to solve
Solve \[-4x+y-5z=2\], \[4x+2y+z=1\], \[2x-y+3z=4\]
Solution
STEP 1
Solve the following system:
{-4 x + y - 5 z = 2
4 x + 2 y + z = 1
2 x - y + 3 z = 4
STEP 2
Express the system in matrix form:
(-4 | 1 | -5
4 | 2 | 1
2 | -1 | 3)(x
y
z) = (2
1
4)
STEP 3
Write the system in augmented matrix form and use Gaussian elimination:
(-4 | 1 | -5 | 2
4 | 2 | 1 | 1
2 | -1 | 3 | 4)
STEP 4
Add row 1 to row 2:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
2 | -1 | 3 | 4)
STEP 5
Add 1/2 × (row 1) to row 3:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
0 | -1/2 | 1/2 | 5)
STEP 6
Multiply row 3 by 2:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
0 | -1 | 1 | 10)
STEP 7
Add 1/3 × (row 2) to row 3:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
0 | 0 | -1/3 | 11)
STEP 8
Multiply row 3 by 3:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
0 | 0 | -1 | 33)
STEP 9
Multiply row 3 by -1:
(-4 | 1 | -5 | 2
0 | 3 | -4 | 3
0 | 0 | 1 | -33)
STEP 10
Add 4 × (row 3) to row 2:
(-4 | 1 | -5 | 2
0 | 3 | 0 | -129
0 | 0 | 1 | -33)
STEP 11
Divide row 2 by 3:
(-4 | 1 | -5 | 2
0 | 1 | 0 | -43
0 | 0 | 1 | -33)
STEP 12
Subtract row 2 from row 1:
(-4 | 0 | -5 | 45
0 | 1 | 0 | -43
0 | 0 | 1 | -33)
STEP 13
Add 5 × (row 3) to row 1:
(-4 | 0 | 0 | -120
0 | 1 | 0 | -43
0 | 0 | 1 | -33)
STEP 14
Divide row 1 by -4:
(1 | 0 | 0 | 30
0 | 1 | 0 | -43
0 | 0 | 1 | -33)
STEP 15
Collect results:
Answer: | 
 | {x = 30
y = -43
z = -33