Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Please help me..
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Sure! I can help you with that. To find the complete solution to the system of equations using Gaussian elimination, we will perform a series of row operations to transform the matrix into echelon form or reduced echelon form.
Step 1: Write the augmented matrix for the system of equations:
| 4 -1 3 | 12 |
| 1 4 6 | -32 |
| 5 3 9 | 20 |
Step 2: Perform row operations to transform the matrix into echelon form. The goal is to create zeros below the main diagonal.
R2 = R2 - ( (1/4) * R1 )
R3 = R3 - (5 * R1 )
The new matrix becomes:
| 4 -1 3 | 12 |
| 0 4 4 | -35 |
| 0 8 -6 | -40 |
Step 3: Perform additional row operations to continue transforming the matrix into echelon form.
R3 = R3 - (2 * R2 )
The new matrix becomes:
| 4 -1 3 | 12 |
| 0 4 4 | -35 |
| 0 0 -14 | -10 |
Step 4: Now, we have our matrix in echelon form. To convert it to reduced echelon form, we will perform the following row operations.
R3 = ( -1/14 ) * R3
The new matrix becomes:
| 4 -1 3 | 12 |
| 0 4 4 | -35 |
| 0 0 1 | 10/14 |
Step 5: Perform further row operations to create zeros above the main diagonal, if necessary.
R2 = R2 - ( (4/1) * R3 )
R1 = R1 - ( (3/1) * R3 )
The new matrix becomes:
| 4 -1 0 | 12 - (3/(14/10)) |
| 0 4 0 | -35 - ( (4/1) * (10/14) ) |
| 0 0 1 | 10/14 |
Simplifying the expressions within the matrix:
| 4 -1 0 | (156 - 45)/14 |
| 0 4 0 | (-490 - 40)/14 |
| 0 0 1 | 10/14 |
This simplifies to:
| 4 -1 0 | 111/14 |
| 0 4 0 | -17 |
| 0 0 1 | 5/7 |
Step 6: Now let's use back substitution to find the solutions. Starting from the last row, solve for z:
z = 5/7
Moving to the second row, solve for y:
4y = -17
y = -17/4
Finally, in the first row, solve for x:
4x - (-17/4) + 0 = 111/14
4x + 17/4 = 111/14
4x = 111/14 - 17/4
4x = (111*2 - 17*7)/(14*4)
4x = 35/4
x = 35/16
So, the complete solution to the system of equations is:
x = 35/16, y = -17/4, z = 5/7