Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. x+3y+6z=6 y-4z=0
type in your equations here and see all the steps:
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
use gaussian elimination to find the complete solution to the system of equations, or show that non exists
5x+13y-24z=8
2x+7y-12z=-1
x+2y-4z=3
To solve the given system of equations using Gaussian elimination, we can follow these steps:
Step 1: Write the augmented matrix for the system of equations:
The augmented matrix can be written as:
[1 3 6 | 6]
[0 1 -4 | 0]
Step 2: Perform row operations to transform the matrix into row-echelon form:
We'll start by dividing the first row by 1 to make the leading entry 1 in the first column:
[1 3 6 | 6]
[0 1 -4 | 0]
Next, we'll perform the row operation R2 = R2 - (-4) * R1 to eliminate the coefficient in the second row, first column:
[1 3 6 | 6]
[0 1 -4 | 0]
Step 3: Continue performing row operations to transform the matrix into reduced row-echelon form:
Next, we can subtract 3 times the second row from the first row to eliminate the coefficient in the first row, second column:
[1 0 18 | 6]
[0 1 -4 | 0]
Now, the matrix is in reduced row-echelon form.
Step 4: Write the system of equations using the reduced row-echelon form:
From the matrix, we can rewrite the system of equations:
x + 0y + 18z = 6
0x + y - 4z = 0
Simplifying the equations, we get:
x + 18z = 6
y - 4z = 0
Step 5: Express the variables in terms of the leading variables:
From the equations, we can express the variables as follows:
x = 6 - 18z
y = 4z
Step 6: Write the complete solution to the system of equations:
The complete solution to the system of equations is:
x = 6 - 18z
y = 4z
z can take any real value.
In summary, the solution to the given system of equations using Gaussian elimination is x = 6 - 18z, y = 4z, where z can be any real number.