Problem to solve
Solve \[4x-3y=7\], \[2x-3y+z=7\], \[4x-y+z=-8\]
Solution
STEP 1
Solve the following system:
{-3 y + 4 x = 7
2 x - 3 y + z = 7
4 x - y + z = -8
STEP 2
Express the system in matrix form:
(4 | -3 | 0
2 | -3 | 1
4 | -1 | 1)(x
y
z) = (7
7
-8)
STEP 3
Write the system in augmented matrix form and use Gaussian elimination:
(4 | -3 | 0 | 7
2 | -3 | 1 | 7
4 | -1 | 1 | -8)
STEP 4
Subtract 1/2 × (row 1) from row 2:
(4 | -3 | 0 | 7
0 | -3/2 | 1 | 7/2
4 | -1 | 1 | -8)
STEP 5
Multiply row 2 by 2:
(4 | -3 | 0 | 7
0 | -3 | 2 | 7
4 | -1 | 1 | -8)
STEP 6
Subtract row 1 from row 3:
(4 | -3 | 0 | 7
0 | -3 | 2 | 7
0 | 2 | 1 | -15)
STEP 7
Add 2/3 × (row 2) to row 3:
(4 | -3 | 0 | 7
0 | -3 | 2 | 7
0 | 0 | 7/3 | -31/3)
STEP 8
Multiply row 3 by 3:
(4 | -3 | 0 | 7
0 | -3 | 2 | 7
0 | 0 | 7 | -31)
STEP 9
Divide row 3 by 7:
(4 | -3 | 0 | 7
0 | -3 | 2 | 7
0 | 0 | 1 | -31/7)
STEP 10
Subtract 2 × (row 3) from row 2:
(4 | -3 | 0 | 7
0 | -3 | 0 | 111/7
0 | 0 | 1 | -31/7)
STEP 11
Divide row 2 by -3:
(4 | -3 | 0 | 7
0 | 1 | 0 | -37/7
0 | 0 | 1 | -31/7)
STEP 12
Add 3 × (row 2) to row 1:
(4 | 0 | 0 | -62/7
0 | 1 | 0 | -37/7
0 | 0 | 1 | -31/7)
STEP 13
Divide row 1 by 4:
(1 | 0 | 0 | -31/14
0 | 1 | 0 | -37/7
0 | 0 | 1 | -31/7)
STEP 14
Collect results:
Answer: | 
 | {x = -31/14
y = -37/7
z = -31/7