Answer :
To find all solutions to the given system of equations using Gauss-Jordan elimination, we can perform row operations to transform the augmented matrix into row-echelon form and then into reduced row-echelon form. The reduced row-echelon form will reveal the solutions of the system and, Hence the system has infinitely many solutions with the variables x1, x2, x3, and x4 expressed in terms of x2 and x4.
We start by constructing the augmented matrix for the system of equations:
[ 0 0 11 13 | 0 ]
[ 7 7 7 7 | 7 ]
[ 16 -7 7 16 | 0 ]
[ 18 -7 7 7 | 0 ]
Next, we perform row operations to transform the matrix into row-echelon form. We can subtract multiples of one row from another to create zeros below the pivots. After row operations, the matrix becomes:
[ 7 7 7 7 | 7 ]
[ 0 -14 0 2 | -7 ]
[ 0 0 11 13 | 0 ]
[ 0 0 14 14 | 0 ]
We continue to perform row operations to further transform the matrix into reduced row-echelon form. After additional row operations, the matrix becomes:
[ 7 0 0 1 | 1 ]
[ 0 -14 0 2 | -7 ]
[ 0 0 11 13 | 0 ]
[ 0 0 0 0 | 0 ]
From the reduced row-echelon form, we can see that the system has infinitely many solutions. The variables can be expressed in terms of the free variables x2 and x4. Therefore, the solution set can be written as:
x1 = 1 - x4
x2 = x2
x3 = 0
x4 = x4
Thus, the system has infinitely many solutions with the variables x1, x2, x3, and x4 expressed in terms of x2 and x4.
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