Question

Solve the linear system by Gaussian elimination.

2x₁+2x₂+2x₃= 0
-2x₁+5x₂+2x₃= 1
8x₁+x₂+4x₃= -1

Answer 1

To solve the linear system by Gaussian elimination, follow these steps: rewrite the system in matrix form, eliminate variables below the main diagonal, write the augmented matrix in row-echelon form, and solve for the variables using back substitution.

Step-by-step explanation:

To solve this linear system by Gaussian elimination, we can use the following steps:

1. Rewrite the system of equations in matrix form:
2. [2 2 2| 0
3. 
   
4. -2 5 2| 1
5. 
   
6. 8 1 4|-1]
7. Perform row operations to eliminate the variables below the main diagonal:
8. [2 2 2| 0
9. 
   
10. 0 3 4| 1
11. 
    
12. 0 -15 -4| -1]
13. Perform additional row operations to write the augmented matrix in row-echelon form:
14. [2 2 2| 0
15. 
    
16. 0 1 4/3| 1/3
17. 
    
18. 0 0 -86/3| -76/3]
19. Solve for the variables using back substitution:
20. x₁ = 6/43, x₂ = 11/43, x₃ = 76/86