Can you check my answer?
If it is wrong can you please explain why and give the correct answer?
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
answer:2x+4y+z=-4
8y-5z=-17
z=1
Your answer gives
x = -1
y = -3/2
z = 1
but these do not satisfy the original equations. I already gave you a url which shows the row-echelon details.
To check your answer, let's go through the steps of Gaussian elimination together.
Step 1: Create an augmented matrix using the coefficients of the variables and the constants on the right-hand side of the equations.
The given system of equations can be represented as follows:
| 2 4 1 | -4 |
| 2 -4 6 | 13 |
| 4 -2 1 | 6 |
Step 2: Perform row operations to simplify the matrix and convert it into row-echelon form.
First, let's focus on the first column. We want the first element of the first row to be a non-zero value. We can achieve this by swapping the first and second row.
| 2 4 1 | -4 |
| 2 -4 6 | 13 |
| 4 -2 1 | 6 |
Next, we need to eliminate the coefficient below the pivot (the first element of the first row), which is 2. Multiply the first row by -1 and add it to the second row.
| 2 4 1 | -4 |
| 0 -8 5 | 17 |
| 4 -2 1 | 6 |
Similarly, we eliminate the coefficient below the pivot in the third row. Multiply the first row by -2 and add it to the third row.
| 2 4 1 | -4 |
| 0 -8 5 | 17 |
| 0 -10 -1 | 14 |
Lastly, we eliminate the coefficient below the pivot in the third row. Multiply the second row by -5/8 and add it to the third row.
| 2 4 1 | -4 |
| 0 -8 5 | 17 |
| 0 0 -9/8 | -11/8 |
Step 3: Simplify the row-echelon matrix and express it as a system of equations.
From the row-echelon form obtained, we can see that the system of equations can be written as:
2x + 4y + z = -4
-8y + 5z = -17
-9/8z = -11/8
However, let's carefully analyze the third equation.
-9/8z = -11/8
To eliminate the fraction, we multiply both sides of the equation by 8:
-9z = -11
Divide both sides by -9 to get:
z = 11/9
Hence, the corrected answer would be:
2x + 4y + z = -4
-8y + 5z = -17
z = 11/9