Use Crammer’s rule to compute the solutions of the following systems.
4x – y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
First we need to set up the determinants for Cramer's Rule:
1. The determinant of the coefficient matrix (D):
D = | 4 -1 2 |
| 1 2 -2 |
| -1 1 1 |
2. The determinant of the x matrix (Dx):
Dx = | 13 -1 2 |
| 0 2 -2 |
| 5 1 1 |
3. The determinant of the y matrix (Dy):
Dy = | 4 13 2 |
| 1 0 -2 |
| -1 5 1 |
4. The determinant of the z matrix (Dz):
Dz = | 4 -1 13 |
| 1 2 0 |
| -1 1 5 |
Now we can calculate each determinant:
- D = 4(2*1 - -2*1) - (-1)(1*1 - -2*-1) + 2(1*1 - 1*2)
= 4(2 + 2) - (-1)(1 + 2) + 2(1 - 2)
= 16 - (-3) + 2(-1)
= 19 + (-2)
= 17
- Dx = 13(2*1 - -2*1) - (-1)(0*1 - -2*5) + 2(0*1 - 5*1)
= 13(2 + 2) - (-1)(0 + 10) + 2(0 - 5)
= 52 - (-10) - 10
= 42 + 10
= 52
- Dy = 4(0*1 - -2*5) - 13(1*1 - -2*1) + 2(1*5 - -1*1)
= 4(0 + 10) - 13(1 + 2) + 2(5 + 1)
= 4(10) - 13(3) + 2(6)
= 40 - 39 + 12
= 13
- Dz = 4(2*5 - 0*1) - (-1)(1*5 - 0*1) + 13(1*1 - 2*1)
= 4(10 - 0) - (-1)(5 - 0) + 13(1 - 2)
= 4(10) - (-5) + 13(-1)
= 40 + 5 - 13
= 32
Now we can find the solutions for x, y, and z:
x = Dx / D = 52 / 17 = 3.0588
y = Dy / D = 13 / 17 = 0.7647
z = Dz / D = 32 / 17 = 1.8824
Therefore, the solutions to the system of equations are: x = 3.0588, y = 0.7647, z = 1.8824.