Answer:
To solve the given system of equations using the Gaussian Elimination method, we'll first write down the augmented matrix and then perform row operations to simplify it.
The given system of equations is:
1x + 1x₂ + 882x₃ = 582 (Equation 1)
2x + 6x₂ + 2x₂ + 8x₃ = -7 (Equation 2)
8x + 2x₂ + 6x₃ = 26 (Equation 3)
We can represent this system as an augmented matrix [A|b]:
[ 1 1 882 | 582 ]
[ 2 6 2 | -7 ]
[ 8 2 6 | 26 ]
Now, we'll perform row operations to simplify the augmented matrix:
R2 = R2 - 2R1
R3 = R3 - 8R1
The updated matrix becomes:
[ 1 1 882 | 582 ]
[ 0 4 -1762 | -1171 ]
[ 0 -6 -6976 | -4562 ]
Next, we'll perform additional row operations to further simplify the matrix:
R2 = R2/4
R3 = R3 + 1.5R2
The updated matrix becomes:
[ 1 1 882 | 582 ]
[ 0 1 -441 | -292.75 ]
[ 0 0 -11132 | -7610.625 ]
Finally, we'll perform the last row operation:
R3 = R3/(-11132)
The final matrix becomes:
[ 1 1 882 | 582 ]
[ 0 1 -441 | -292.75 ]
[ 0 0 1 | 0.68475 ]
Now, we'll work our way back up, starting from the bottom row, to find the values of the variables:
From the third row, we can determine that x₃ = 0.68475.
Substituting this value back into the second row, we can solve for x₂:
1x + 1(-441) + 882(0.68475) = 582
x - 441 + 604.05 = 582
x - 441 = -22.05
x = -22.05 + 441
x = 418.95
Finally, substituting the values of x₃ = 0.68475 and x₂ = 418.95 into the first row, we can solve for x:
1(418.95) + 1(-441) + 882(0.68475) = 582
418.95 - 441 + 604.05 = 582
582 = 582
Therefore, the solution to the given system of equations is:
x = 418.95
x₂ = -441
x₃ = 0.68475