First, we need to find the determinant of the coefficient matrix:
| 4 -1 2 |
| 1 2 -2 |
| -1 1 1 |
Det = 4(2)(1) + 1(-2)(-1) + (-1)(1)(2) - (-1)(2)(1) - 1(-1)(4) - 4(1)(-2)
Det = 8 + 2 + 2 - 2 + 4 + 8
Det = 22
Next, we need to find the determinants for x, y, and z:
For x:
| 13 -1 2 |
| 0 2 -2 |
| 5 1 1 |
Det_x = 13(2)(1) + (-1)(-2)(5) + 2(1)(5) - 5(2)(2) - (-1)(1)(13) - 13(1)(-2)
Det_x = 26 + 10 + 10 - 20 + 13 + 26
Det_x = 65
For y:
| 4 13 2 |
| 1 0 -2 |
| -1 5 1 |
Det_y = 4(0)(1) + 13(-2)(-1) + 2(5)(-1) - (-1)(0)(2) - 1(5)(4) - 4(1)(-2)
Det_y = 0 + 26 - 10 - 0 - 20 + 8
Det_y = 4
For z:
| 4 -1 13 |
| 1 2 0 |
| -1 1 5 |
Det_z = 4(2)(5) + (-1)(0)(-1) + 13(1)(1) - (-1)(2)(13) - 1(1)(4) - 4(0)(5)
Det_z = 40 + 0 + 13 + 26 - 4 + 0
Det_z = 75
Finally, we can find the solutions:
x = Det_x / Det = 65 / 22 = 2.9545
y = Det_y / Det = 4 / 22 = 0.1818
z = Det_z / Det = 75 / 22 = 3.4091
So, the solutions to the system of equations are x ≈ 2.9545, y ≈ 0.1818, z ≈ 3.4091.
Use Crammer’s rule to compute the solutions of the following systems.
4x – y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
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