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
4 -1 3 12
1 4 6 -32
5 3 9 20
I believe that the solution does not exist and how can I prove that?
never mind the matrix stuff.
Suppose we subtract #3 - #1
I get x + 4y + 6z = 8
but #2 says:
x + 4y + 6z = -32
which contradicts the above, so ....
no solution
To solve the system of equations using Gaussian elimination, we will need to perform row operations on the augmented matrix until it is in row-echelon form. If we encounter a row of all zeros on the left-hand side but a non-zero value on the right-hand side, it implies that there is no solution.
Let's perform the row operations step by step:
Step 1: Swap the first row with the second row (to begin with a leading coefficient of 1).
1 4 6 -32
4 -1 3 12
5 3 9 20
Step 2: Multiply the first row by 4 and subtract it from the second row to eliminate the first variable in the second row.
1 4 6 -32
0 -17 -21 140
5 3 9 20
Step 3: Multiply the first row by 5 and subtract it from the third row to eliminate the first variable in the third row.
1 4 6 -32
0 -17 -21 140
0 -17 -21 180
Step 4: Divide the second row by -17 to make the leading coefficient of the second row 1.
1 4 6 -32
0 1 1 -8.24
0 -17 -21 180
Step 5: Multiply the second row by 4 and subtract it from the first row to eliminate the second variable in the first row.
1 0 2 -0.76
0 1 1 -8.24
0 -17 -21 180
Step 6: Multiply the second row by 17 and add it to the third row to eliminate the second variable in the third row.
1 0 2 -0.76
0 1 1 -8.24
0 0 -4 -.16
Step 7: Divide the third row by -4 to make the leading coefficient of the third row 1.
1 0 2 -0.76
0 1 1 -8.24
0 0 1 0.04
Now, we have the matrix in row-echelon form. Let's work backward to find the solution:
From the third row, we get:
z = 0.04
Substituting this value back into the second row, we get:
y + 0.04 = -8.24
Simplifying the equation, we find:
y = -8.28
Substituting the values of y and z into the first row, we get:
x + 2(-8.28) - 0.76 = -0.76
Simplifying the equation, we find:
x = -0.68
So, the complete solution to the system of equations is:
x = -0.68, y = -8.28, z = 0.04
Therefore, the solution does exist, and it is not empty.