Use Gaussian elimination to write the system of equations given below in row-echelon form
2x+4y+z=-4
2x-4y+6z=13
4x-2y+z=6
see all the details here:
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
To solve this system of equations using Gaussian elimination, follow these steps:
Step 1: Write the system of equations in augmented matrix form.
The augmented matrix for the given system of equations is:
| 2 4 1 | -4 |
| 2 -4 6 | 13 |
| 4 -2 1 | 6 |
Step 2: Perform row operations to transform the matrix into row-echelon form.
Start by making the element in the (1,1) position equal to 1 by dividing the first row by 2:
| 1 2 0.5 | -2 |
| 2 -4 6 | 13 |
| 4 -2 1 | 6 |
Next, eliminate the 2 below the leading 1 in the second row by subtracting 2 times the first row from the second row:
| 1 2 0.5 | -2 |
| 0 -8 5 | 17 |
| 4 -2 1 | 6 |
Now, eliminate the 4 below the leading 1 in the third row by subtracting 4 times the first row from the third row:
| 1 2 0.5 | -2 |
| 0 -8 5 | 17 |
| 0 -10 -0.5 | 14 |
Finally, eliminate the -10 below the leading -8 in the third row by adding 1.25 times the second row to the third row:
| 1 2 0.5 | -2 |
| 0 -8 5 | 17 |
| 0 0 0 | 14.25 |
Step 3: Write the system of equations in row-echelon form.
The row-echelon form of the augmented matrix corresponds to the following system of equations:
1x + 2y + 0.5z = -2
0x - 8y + 5z = 17
0x + 0y + 0z = 14.25
Step 4: Solve the system of equations.
From the last equation, we can see that 0 = 14.25, which is not possible. Therefore, the system of equations is inconsistent, meaning there is no solution.
In row-echelon form, the system of equations represents the same relationships as the original system, but in a simplified and easier-to-analyze form.