Use the echelon method to solve the system:
x + 3y + 4z = 14
2x – 3y +2z = 10
3x – 9y + z = 9
To solve the given system of linear equations using the echelon method, we will perform a series of row operations to bring the system to its row-echelon form.
Step 1: Write down the augmented matrix of the system:
[1 3 4 | 14]
[2 -3 2 | 10]
[3 -9 1 | 9]
Step 2: Using row operations, we will aim to create zeros below the pivot elements. Start by creating zeros in the elements below the first pivot element, which is 1.
- Row 2 = Row 2 - 2 * Row 1
- Row 3 = Row 3 - 3 * Row 1
The updated matrix is:
[1 3 4 | 14]
[0 -9 -6 | -18]
[0 -18 -11 | -33]
Step 3: Proceed to the second row and make the second pivot element, which is -9, equal to 1 by multiplying the row by -1/9.
- Row 2 = (-1/9) * Row 2
The updated matrix becomes:
[1 3 4 | 14]
[0 1 2/3 | 2]
[0 -18 -11 | -33]
Step 4: Now, we will aim for a zero below the second pivot element in the third row.
- Row 3 = Row 3 + 18 * Row 2
The updated matrix is:
[1 3 4 | 14]
[0 1 2/3 | 2]
[0 0 1 | -9]
Step 5: The matrix is now in row-echelon form. To find the solution, we will work our way back up from the last row.
- z = -9
Substitute this value of z into the second equation:
- y + (2/3)(-9) = 2
- y - 6 = 2
- y = 8
Finally, substitute the values of y and z into the first equation:
x + 3(8) + 4(-9) = 14
x + 24 - 36 = 14
x - 12 = 14
x = 26
Therefore, the solution to the system of equations is x = 26, y = 8, and z = -9.