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Use the​ Gauss-Jordan method to solve the following system of equations. 2x + 3y - z = 0 5x - y + 3x = 1 7x + 2y + 2z = 1

User BachT
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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.

User Trejder
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