Final answer:
To solve the given system of equations using Cramer's Rule, we calculate the determinant of the coefficient matrix (D) and then find the determinants Dx, Dy, and Dz after replacing their respective columns with the constant terms. Using these determinants, we find the values of x, y, and z by dividing Dx, Dy, and Dz by D.
Step-by-step explanation:
To solve the system of equations using Cramer's Rule, we'll use determinants. The given system is:
- 2x - z = 14
- 3x - y + 5z = 0
- 4x + 2y + 3z = -2
First, we find the determinant of the coefficient matrix (D), which includes the coefficients of x, y, and z from each equation:
D = |
2 0 -1 |
3 -1 5 |
4 2 3 |
Now, we'll find the determinants for Dx, Dy, and Dz by replacing the respective columns with the constants from the right side of the equations:
Dx = |
14 0 -1 |
0 -1 5 |
-2 2 3 |
Dy = |
2 14 -1 |
3 0 5 |
4 -2 3 |
Dz = |
2 0 14 |
3 -1 0 |
4 2 -2 |
Calculating these determinants, we can then find the value of x, y, and z:
x = Dx / D
y = Dy / D
z = Dz / D
Finally, we substitute the values for Dx, Dy, and Dz, and solve for the values of x, y, and z.