Problem to solve
Solve \[-10x+3y+z=-2\], \[3x-2y+5z=4\], \[-6x+4y+2z=1\]
Solution
STEP 1
Solve the following system:
{-10 x + 3 y + z = -2
3 x - 2 y + 5 z = 4
-6 x + 4 y + 2 z = 1
STEP 2
Express the system in matrix form:
(-10 | 3 | 1
3 | -2 | 5
-6 | 4 | 2)(x
y
z) = (-2
4
1)
STEP 3
Solve the system with Cramer's rule:
x = -2 | 3 | 1
4 | -2 | 5
1 | 4 | 2/-10 | 3 | 1
3 | -2 | 5
-6 | 4 | 2 and y = -10 | -2 | 1
3 | 4 | 5
-6 | 1 | 2/-10 | 3 | 1
3 | -2 | 5
-6 | 4 | 2 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/-10 | 3 | 1
3 | -2 | 5
-6 | 4 | 2
STEP 4
Evaluate the determinant -10 | 3 | 1
3 | -2 | 5
-6 | 4 | 2 = 132:
x = -2 | 3 | 1
4 | -2 | 5
1 | 4 | 2/132 and y = -10 | -2 | 1
3 | 4 | 5
-6 | 1 | 2/132 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/132
STEP 5
Evaluate the determinant -2 | 3 | 1
4 | -2 | 5
1 | 4 | 2 = 57:
x = 57/132 and y = -10 | -2 | 1
3 | 4 | 5
-6 | 1 | 2/132 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/132
STEP 6
The gcd of 57 and 132 is 3, so 57/132 = (19×3)/(44×3) = 19/44:
x = 19/44 and y = -10 | -2 | 1
3 | 4 | 5
-6 | 1 | 2/132 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/132
STEP 7
Evaluate the determinant -10 | -2 | 1
3 | 4 | 5
-6 | 1 | 2 = 69:
x = 19/44 and y = 69/132 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/132
STEP 8
The gcd of 69 and 132 is 3, so 69/132 = (23×3)/(44×3) = 23/44:
x = 19/44 and y = 23/44 and z = -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1/132
STEP 9
Evaluate the determinant -10 | 3 | -2
3 | -2 | 4
-6 | 4 | 1 = 99:
x = 19/44 and y = 23/44 and z = 99/132
STEP 10
The gcd of 99 and 132 is 33, so 99/132 = (3×33)/(4×33) = 3/4:
Answer: | 
 | x = 19/44 and y = 23/44 and z = 3/4