Use the Gauss-Jordan method to solve the equation:
3x-5y-2z=-9
-4x+3y+z=11
8x-5y+4z=6
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To solve the given system of equations using the Gauss-Jordan method, we will perform row operations on an augmented matrix.
Step 1: Create the augmented matrix
Start by writing down the coefficients of the variables and the constants in a matrix form:
[ 3 -5 -2 | -9 ]
[ -4 3 1 | 11 ]
[ 8 -5 4 | 6 ]
Step 2: Perform row operations to transform the augmented matrix into row-echelon form
The goal of row operations is to create zeros below the main diagonal. To achieve this, we can use elementary row operations such as adding or subtracting multiples of one row to/from another, and multiplying a row by a scalar.
a) Multiply Row 1 by -4 and add it to Row 2:
[ 3 -5 -2 | -9 ]
[ 0 -17 -7 | 35 ]
[ 8 -5 4 | 6 ]
b) Multiply Row 1 by 8 and subtract it from Row 3:
[ 3 -5 -2 | -9 ]
[ 0 -17 -7 | 35 ]
[ 0 35 20 | 78 ]
Step 3: Continue row operations to convert the augmented matrix into reduced row-echelon form
To create zeros above the main diagonal, we will perform additional row operations.
a) Multiply Row 2 by -2 and add it to Row 1:
[ 3 -5 -2 | -9 ]
[ 0 -17 -7 | 35 ]
[ 0 35 20 | 78 ]
b) Multiply Row 3 by 3 and add it to Row 2:
[ 3 -5 -2 | -9 ]
[ 0 -17 -7 | 35 ]
[ 0 0 41 | 269 ]
Step 4: Divide each row by the leading coefficient to get leading ones
In this case, we need to divide Row 3 by 41 to obtain a leading one:
[ 3 -5 -2 | -9 ]
[ 0 -17 -7 | 35 ]
[ 0 0 1 | 6.56 ]
Step 5: Perform back substitution
Using the reduced row-echelon form of the augmented matrix, we can now write the system of equations:
3x - 5y - 2z = -9 (equation 1)
-17y - 7z = 35 (equation 2)
z = 6.56 (equation 3)
From equation 3, we can find that z = 6.56.
Substituting this value of z into equation 2, we get:
-17y - 7(6.56) = 35
Simplifying this equation, we find that -17y - 45.92 = 35.
Solving for y, we get y = -2.16.
Finally, substitute the values of y and z into equation 1 to find x:
3x - 5(-2.16) - 2(6.56) = -9
Simplifying this equation, we find that 3x + 10.8 - 13.12 = -9.
Solving for x, we get x = -1.2.
Therefore, the solution to the given system of equations is x = -1.2, y = -2.16, and z = 6.56.