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
Solve the system with Cramer's rule:
x = 2 | 1 | -5
1 | 2 | 1
4 | -1 | 3/-4 | 1 | -5
4 | 2 | 1
2 | -1 | 3 and y = -4 | 2 | -5
4 | 1 | 1
2 | 4 | 3/-4 | 1 | -5
4 | 2 | 1
2 | -1 | 3 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/-4 | 1 | -5
4 | 2 | 1
2 | -1 | 3
STEP 4
Evaluate the determinant -4 | 1 | -5
4 | 2 | 1
2 | -1 | 3 = 2:
x = 2 | 1 | -5
1 | 2 | 1
4 | -1 | 3/2 and y = -4 | 2 | -5
4 | 1 | 1
2 | 4 | 3/2 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/2
STEP 5
Evaluate the determinant 2 | 1 | -5
1 | 2 | 1
4 | -1 | 3 = 60:
x = 60/2 and y = -4 | 2 | -5
4 | 1 | 1
2 | 4 | 3/2 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/2
STEP 6
Divide 60 by 2: 60/2 = (30×2)/(1×2) = 30:
x = 30 and y = -4 | 2 | -5
4 | 1 | 1
2 | 4 | 3/2 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/2
STEP 7
Evaluate the determinant -4 | 2 | -5
4 | 1 | 1
2 | 4 | 3 = -86:
x = 30 and y = (-86)/2 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/2
STEP 8
Divide -86 by 2: (-86)/2 = (-43×2)/(1×2) = -43:
x = 30 and y = -43 and z = -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4/2
STEP 9
Evaluate the determinant -4 | 1 | 2
4 | 2 | 1
2 | -1 | 4 = -66:
x = 30 and y = -43 and z = (-66)/2
STEP 10
Divide -66 by 2: (-66)/2 = (-33×2)/(1×2) = -33:
Answer: | 
 | x = 30 and y = -43 and z = -33