Introduction
To solve the system of equations using Cramer's Rule, we first express the given equations in standard form:
1. 4x - 5y - 2z = 0
2. 2x + 2y + z = 2
3. 2x + 2y + 8z = -1
Step 1: Form the Coefficient Matrix (D)
The coefficient matrix D is formed from the coefficients of x, y, and z:
- D = | 4 -5 -2 |
- | 2 2 1 |
- | 2 2 8 |
Step 2: Calculate the Determinant (D)
To find the determinant D, apply the rule of Sarrus or the method of minors:
- D = 4(2*8 - 1*2) - (-5)(2*8 - 1*2) - 2(2*2 - 2*2)
Calculating this gives D = 4(16 - 2) + 5(16 - 2) - 2(0) = 4*14 + 5*14 = 56 + 70 = 126.
Step 3: Form Matrices for x, y, z
Next, create matrices Dx, Dy, and Dz by replacing the first, second, and third columns of D with the constants from the right side of the equations.
- Dx = | 0 -5 -2 |
- | 2 2 1 |
- | -1 2 8 |
- Dy = | 4 0 -2 |
- | 2 2 1 |
- | 2 -1 8 |
- Dz = | 4 -5 0 |
- | 2 2 2 |
- | 2 2 -1 |
Step 4: Calculate Determinants (Dx, Dy, Dz)
Calculate the determinants of Dx, Dy, and Dz using the same method as D.
Step 5: Solve for x, y, z
Using Cramer’s Rule:
- x = Dx/D
- y = Dy/D
- z = Dz/D
This will yield the values of x, y, and z after appropriate calculations.
Conclusion
Following these steps meticulously will lead to the solutions for x, y, and z using Cramer’s Rule.