Use the echelon Method to solve the system of three equations in three unknowns.
2x+y+z=9
-x-y+z=1
3x-y+z=9
To solve the system of three equations in three unknowns using the echelon method, follow these steps:
Step 1: Write the augmented matrix for the system of equations.
The augmented matrix is a matrix that includes the coefficients of the variables and the constants in the system of equations. For the given system, the augmented matrix is:
| 2 1 1 | 9 |
| -1 -1 1 | 1 |
| 3 -1 1 | 9 |
Step 2: Perform row operations to transform the matrix into echelon form.
The goal is to transform the matrix into a triangular form with leading 1's in each row. To do this, perform row operations such as swapping rows, multiplying and adding rows. The objective is to eliminate the coefficients below the leading entries.
Start with the first column and use row operations to create zeros below the leading entry of the first row.
Row 2 = Row 2 + (1/2) * Row 1 -> This eliminates the -1 below the leading 2.
| 2 1 1 | 9 |
| 0 0.5 1.5 | 5.5 |
| 3 -1 1 | 9 |
Row 3 = Row 3 - (3/2) * Row 1 -> This eliminates the 3 below the leading 2.
| 2 1 1 | 9 |
| 0 0.5 1.5 | 5.5 |
| 0 -2.5 -0.5 | -7.5 |
Next, work on the second column to make the coefficient below the leading entry of the second row zero.
Row 3 = Row 3 + (5/2) * Row 2 -> This eliminates the -2.5 below the leading 0.5.
| 2 1 1 | 9 |
| 0 0.5 1.5 | 5.5 |
| 0 0 4 | 11 |
Step 3: Transform the echelon form into reduced echelon form.
To obtain the reduced echelon form, make the leading entry in each row 1 and eliminate any non-zero coefficients above the leading entry.
Row 2 = Row 2 / 0.5 -> This makes the leading entry of the second row 1.
| 2 1 1 | 9 |
| 0 1 3 | 11 |
| 0 0 4 | 11 |
Row 1 = Row 1 - Row 2 -> This eliminates the coefficient below the leading 1 of the first row.
| 2 0 -2 | -2 |
| 0 1 3 | 11 |
| 0 0 4 | 11 |
Row 3 = Row 3 / 4 -> This makes the leading entry of the third row 1.
| 2 0 -2 | -2 |
| 0 1 3 | 11 |
| 0 0 1 | 11/4 |
Step 4: Read the solution from the reduced echelon form.
The reduced echelon form tells us the values of the variables. The last column of the matrix represents the constants in the system of equations.
Therefore, the solution to the system of equations is:
x = -2
y = 11
z = 11/4
Thus, the system has a unique solution.
Note: The echelon method can be applied to larger systems of equations by following the same steps. The key is to perform row operations to transform the matrix into echelon form, and then reduce it further to obtain the reduced echelon form.