Problem to solve
Solve \[\frac{x}{2}-3z=-2\], \[3y-\frac{z}{2}=2\], \[x+3y-z=5\]
Solution
STEP 1
Solve the following system:
{-3 z + x/2 = -2
-z/2 + 3 y = 2
x + 3 y - z = 5
STEP 2
Express the system in matrix form:
(1/2 | 0 | -3
0 | 3 | -1/2
1 | 3 | -1)(x
y
z) = (-2
2
5)
STEP 3
Solve the system with Cramer's rule:
x = (-2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/(1/2 | 0 | -3
0 | 3 | -1/2
1 | 3 | -1) and y = (1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/(1/2 | 0 | -3
0 | 3 | -1/2
1 | 3 | -1) and z = (1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(1/2 | 0 | -3
0 | 3 | -1/2
1 | 3 | -1)
STEP 4
Evaluate the determinant 1/2 | 0 | -3
0 | 3 | -1/2
1 | 3 | -1 = 33/4:
x = (-2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/(33/4) and y = (1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/(33/4) and z = (1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(33/4)
STEP 5
The gcd of -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1 and 33/4 is 1/4, so (-2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/(33/4) = (4 -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/(4×33/4) = 4/33 -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1:
x = 4/33 -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1 and y = (1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/(33/4) and z = (1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(33/4)
STEP 6
The gcd of 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1 and 33/4 is 1/4, so (1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/(33/4) = (4 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/(4×33/4) = 4/33 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1:
x = (4 -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/33 and y = 4/33 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1 and z = (1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(33/4)
STEP 7
The gcd of 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5 and 33/4 is 1/4, so (1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(33/4) = (4 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/(4×33/4) = 4/33 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5:
x = (4 -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1)/33 and y = (4 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/33 and z = 4/33 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5
STEP 8
Evaluate the determinant -2 | 0 | -3
2 | 3 | -1/2
5 | 3 | -1 = 30:
x = 4/33×30 and y = (4 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/33 and z = (4 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/33
STEP 9
4/33×30 = 40/11:
x = 40/11 and y = (4 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1)/33 and z = (4 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/33
STEP 10
Evaluate the determinant 1/2 | -2 | -3
0 | 2 | -1/2
1 | 5 | -1 = 29/4:
x = 40/11 and y = 4/33×29/4 and z = (4 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/33
STEP 11
4/33×29/4 = 29/33:
x = 40/11 and y = 29/33 and z = (4 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5)/33
STEP 12
Evaluate the determinant 1/2 | 0 | -2
0 | 3 | 2
1 | 3 | 5 = 21/2:
x = 40/11 and y = 29/33 and z = 4/33×21/2
STEP 13
4/33×21/2 = 14/11:
Answer: | 
 | x = 40/11 and y = 29/33 and z = 14/11