Answer:
Since the third equation is inconsistent (0 ≠ 9/8), there is no solution to the system of equations
Explanation:
To solve the given system of equations using the Gauss-Jordan elimination method, we'll first write the augmented matrix for the system and then perform row operations to bring it to row-echelon form and then reduced row-echelon form.
The system of equations is:
2x + 3y - z = 0
5x - y + 3z = 1
7x + 2y + 2z = 1
Now, let's create the augmented matrix [A|B], where A represents the coefficients of the variables (x, y, z), and B represents the constants on the right side of the equations:
[ 2 3 -1 | 0 ]
[ 5 -1 3 | 1 ]
[ 7 2 2 | 1 ]
Our goal is to transform this matrix into reduced row-echelon form (RREF). We'll use row operations to do that:
Divide Row 1 by 2:
[ 1 3/2 -1/2 | 0 ]
[ 5 -1 3 | 1 ]
[ 7 2 2 | 1 ]
Replace Row 2 with Row 2 - 5 * Row 1:
[ 1 3/2 -1/2 | 0 ]
[ 0 -8/2 5/2 | 1 ]
[ 7 2 2 | 1 ]
Replace Row 3 with Row 3 - 7 * Row 1:
[ 1 3/2 -1/2 | 0 ]
[ 0 -8/2 5/2 | 1 ]
[ 0 -17/2 9/2 | 1 ]
Divide Row 2 by (-4):
[ 1 3/2 -1/2 | 0 ]
[ 0 4 -5 | -1/4 ]
[ 0 -17/2 9/2 | 1 ]
Replace Row 3 with Row 3 + (17/2) * Row 2:
[ 1 3/2 -1/2 | 0 ]
[ 0 4 -5 | -1/4 ]
[ 0 0 0 | 9/8 ]
Now, we have the matrix in row-echelon form. To get it into reduced row-echelon form, we'll continue with row operations:
Divide Row 2 by 4:
[ 1 3/2 -1/2 | 0 ]
[ 0 1 -5/4 | -1/16 ]
[ 0 0 0 | 9/8 ]
Replace Row 1 with Row 1 - (3/2) * Row 2:
[ 1 0 1/8 | 3/32 ]
[ 0 1 -5/4 | -1/16 ]
[ 0 0 0 | 9/8 ]
Now, we have the matrix in reduced row-echelon form. We can interpret this as a system of equations:
x + (1/8)z = 3/32
y - (5/4)z = -1/16
0 = 9/8
Since the third equation is inconsistent (0 ≠ 9/8), there is no solution to the system of equations. The system is inconsistent, and there are no values of x, y, and z that satisfy all three equations simultaneously.