Problem to solve
Solve \[\frac{2x}{3}-2z=-2\], \[2y-4z=2\], \[\frac{x}{3}-5y=2\]
Solution
STEP 1
Solve the following system:
{-2 z + (2 x)/3 = -2
-4 z + 2 y = 2
-5 y + x/3 = 2
STEP 2
Express the system in matrix form:
(2/3 | 0 | -2
0 | 2 | -4
1/3 | -5 | 0)(x
y
z) = (-2
2
2)
STEP 3
Write the system in augmented matrix form and use Gaussian elimination:
(2/3 | 0 | -2 | -2
0 | 2 | -4 | 2
1/3 | -5 | 0 | 2)
STEP 4
Multiply row 1 by 3/2:
(1 | 0 | -3 | -3
0 | 2 | -4 | 2
1/3 | -5 | 0 | 2)
STEP 5
Divide row 2 by 2:
(1 | 0 | -3 | -3
0 | 1 | -2 | 1
1/3 | -5 | 0 | 2)
STEP 6
Multiply row 3 by 3:
(1 | 0 | -3 | -3
0 | 1 | -2 | 1
1 | -15 | 0 | 6)
STEP 7
Subtract row 1 from row 3:
(1 | 0 | -3 | -3
0 | 1 | -2 | 1
0 | -15 | 3 | 9)
STEP 8
Divide row 3 by 3:
(1 | 0 | -3 | -3
0 | 1 | -2 | 1
0 | -5 | 1 | 3)
STEP 9
Swap row 2 with row 3:
(1 | 0 | -3 | -3
0 | -5 | 1 | 3
0 | 1 | -2 | 1)
STEP 10
Add 1/5 × (row 2) to row 3:
(1 | 0 | -3 | -3
0 | -5 | 1 | 3
0 | 0 | -9/5 | 8/5)
STEP 11
Multiply row 3 by 5:
(1 | 0 | -3 | -3
0 | -5 | 1 | 3
0 | 0 | -9 | 8)
STEP 12
Divide row 3 by -9:
(1 | 0 | -3 | -3
0 | -5 | 1 | 3
0 | 0 | 1 | -8/9)
STEP 13
Subtract row 3 from row 2:
(1 | 0 | -3 | -3
0 | -5 | 0 | 35/9
0 | 0 | 1 | -8/9)
STEP 14
Divide row 2 by -5:
(1 | 0 | -3 | -3
0 | 1 | 0 | -7/9
0 | 0 | 1 | -8/9)
STEP 15
Add 3 × (row 3) to row 1:
(1 | 0 | 0 | -17/3
0 | 1 | 0 | -7/9
0 | 0 | 1 | -8/9)
STEP 16
Collect results:
Answer: | 
 | {x = -17/3
y = -7/9
z = -8/9