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
Solve the system with Cramer's rule:
x = -2 | 0 | -2
2 | 2 | -4
2 | -5 | 0/2/3 | 0 | -2
0 | 2 | -4
1/3 | -5 | 0 and y = (2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/(2/3 | 0 | -2
0 | 2 | -4
1/3 | -5 | 0) and z = (2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/(2/3 | 0 | -2
0 | 2 | -4
1/3 | -5 | 0)
STEP 4
Evaluate the determinant 2/3 | 0 | -2
0 | 2 | -4
1/3 | -5 | 0 = -12:
x = -2 | 0 | -2
2 | 2 | -4
2 | -5 | 0/(-12) and y = (2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/(-12) and z = (2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/(-12)
STEP 5
Simplify -2 | 0 | -2
2 | 2 | -4
2 | -5 | 0/(-12):
x = -1/12 -2 | 0 | -2
2 | 2 | -4
2 | -5 | 0 and y = (2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/(-12) and z = (2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/(-12)
STEP 6
Simplify (2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/(-12):
x = -(-2 | 0 | -2
2 | 2 | -4
2 | -5 | 0)/12 and y = -1/12 2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0 and z = (2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/(-12)
STEP 7
Simplify (2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/(-12):
x = -(-2 | 0 | -2
2 | 2 | -4
2 | -5 | 0)/12 and y = -(2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/12 and z = -1/12 2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2
STEP 8
Evaluate the determinant -2 | 0 | -2
2 | 2 | -4
2 | -5 | 0 = 68:
x = (-1)/12×68 and y = -(2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/12 and z = -(2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/12
STEP 9
(-1)/12×68 = (-17)/3:
x = -17/3 and y = -(2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0)/12 and z = -(2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/12
STEP 10
Evaluate the determinant 2/3 | -2 | -2
0 | 2 | -4
1/3 | 2 | 0 = 28/3:
x = (-17)/3 and y = (-1)/12×28/3 and z = -(2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/12
STEP 11
(-1)/12×28/3 = (-7)/9:
x = (-17)/3 and y = -7/9 and z = -(2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2)/12
STEP 12
Evaluate the determinant 2/3 | 0 | -2
0 | 2 | 2
1/3 | -5 | 2 = 32/3:
x = (-17)/3 and y = (-7)/9 and z = (-1)/12×32/3
STEP 13
(-1)/12×32/3 = (-8)/9:
Answer: | 
 | x = (-17)/3 and y = (-7)/9 and z = -8/9