metrix method solve equation 2x-y-3z=9 ,x+y+z=6 ,x-y+z=2
by matrix method, do you mean row operations, or multiplication by the inverse?
To solve this system of equations using the matrix method, we will represent the coefficients and constants of the equations in a matrix form.
Step 1: Set up the augmented matrix
We write down the coefficients and constants of the system in a matrix format, with the variables ordered in columns and the equations as rows:
[ 2 -1 -3 | 9 ]
[ 1 1 1 | 6 ]
[ 1 -1 1 | 2 ]
Step 2: Perform row operations to transform the matrix to row-echelon form
We will use row operations to eliminate variables one at a time and bring the matrix to row-echelon form. In row-echelon form, every leading coefficient is 1, and all entries below and above the leading coefficient are zeros.
First, we need to make the leading coefficient in the first row (2) equal to 1. To do this, we divide the first row by 2:
[ 1 -1.5 -1.5 | 4.5 ]
[ 1 1 1 | 6 ]
[ 1 -1 1 | 2 ]
Next, we want to eliminate the coefficients below the leading coefficient in the first column. To do this, we subtract the first row from the second and third rows:
[ 1 -1.5 -1.5 | 4.5 ]
[ 0 2.5 2.5 | 1.5 ]
[ 0 0 0 | -2.5 ]
Step 3: Continue performing row operations until the matrix is in reduced row-echelon form
Now, we can simplify the system by making the coefficient of the second equation equal to 1. We divide the second row by 2.5:
[ 1 -1.5 -1.5 | 4.5 ]
[ 0 1 1 | 0.6 ]
[ 0 0 0 | -2.5 ]
At this point, we can see that the third equation is inconsistent, as it can't be satisfied. This means the system of equations has no solution.
If the third equation were consistent and solvable, we would continue performing row operations until the matrix is in reduced row-echelon form, which would look like:
[ 1 0 0 | 2 ]
[ 0 1 1 | 0.6 ]
[ 0 0 0 | -2.5 ]
Step 4: Interpret the results
In this case, since the third equation is inconsistent, the system of equations has no solution. This means the system of equations is inconsistent, and the given equations do not have a common solution for x, y, and z.