To use Cramer's rule, we first need to find the determinants of the matrix formed by the coefficients of the variables.
The determinant of the main matrix is:
D = | 4 -1 2 |
| 1 2 -2 |
|-1 1 1 |
Next, we find the determinant of the x matrix by replacing the coefficients of x with the constants:
Dx = | 13 -1 2 |
0 2 -2 |
5 1 1 |
Then, we find the determinant of the y matrix by replacing the coefficients of y with the constants:
Dy = | 4 13 2 |
1 0 -2 |
-1 5 1 |
Finally, we find the determinant of the z matrix by replacing the coefficients of z with the constants:
Dz = | 4 -1 13 |
1 2 0 |
-1 1 5 |
Now, we calculate the determinants:
D = 4(4 - 2) - (-1)(-1) = 17
Dx = 13(4 - 4) - (-1)(-2) = 2
Dy = 4(0 - 10) - 1(5 - 1) = -40
Dz = 4(0 - 8) - 1(3 - 5) = 7
Finally, we obtain the solutions for x, y, and z using Cramer's rule:
x = Dx / D = 2 / 17
y = Dy / D = -40 / 17
z = Dz / D = 7 / 17
Therefore, the solutions for x, y, and z are:
x = 2 / 17
y = -40 / 17
z = 7 / 17
Use Crammer’s rule to solve this equation
4x – y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
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