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Explain Bot To solve the system of linear equations using the Gauss-Jordan method, we need to perform row operations on the augmented matrix until it is in the reduced row-echelon form. Each row operation corresponds to an algebraic operation on the system of equations.
The augmented matrix for the given system of equations is:
[3 1 1 1 | 0]
[5 -1 1 -1 | 0]
Step 1: Swap rows if necessary to get a non-zero entry at the top of the first column.
Looking at the first column, there is already a non-zero entry, so we don't need to swap rows.
Step 2: Use row operations to create zeros below the pivot in the first column.
To create a zero below the pivot (3), we can subtract 5/3 times the first row from the second row.
[3 1 1 1 | 0]
[0 -8/3 -2/3 -8/3 | 0]
Step 3: Divide the first row by the pivot to make it equal to 1.
Dividing the first row by 3, we get:
[1 1/3 1/3 1/3 | 0]
[0 -8/3 -2/3 -8/3 | 0]
Step 4: Use row operations to create zeros above and below the second pivot (if there is one).
To create a zero above the second pivot, we can add 8/9 times the second row to the first row.
[1 0 -1/9 -1 | 0]
[0 -8/3 -2/3 -8/3 | 0]
Step 5: Divide the second row by the second pivot to make it equal to 1.
Dividing the second row by -8/3, we get:
[1 0 -1/9 -1 | 0]
[0 1 1/4 1 | 0]
Step 6: Use row operations to create zeros above the third pivot (if there is one).
Since there is no more pivot to eliminate, we have obtained the reduced row-echelon form of the augmented matrix:
[1 0 -1/9 -1 | 0]
[0 1 1/4 1 | 0]
The solution to the system of equations is:
x1 = -1/9
x2 = -1/4
x3 = 1/9
x4 = 1