Convert the given system of equations to matrix form
2. use row operations to put the matrix in echelon form.
3. find the solutions set and put in vector form.
x+y+w+z-3u=5
x-y-2w+z+2u=4 2x+0+w-z+u=3

The matrix form of the system of equations is

The reduced row echelon form is

The vector form of the general solution for this system is

Step-by-step explanation:

Convert the given system of equations to matrix form

We have the following system of linear equations:

To arrange this system in matrix form (Ax = b), we need the coefficient matrix (A), the variable matrix (x), and the constant matrix (b).

so

Use row operations to put the augmented matrix in echelon form.

An augmented matrix for a system of equations is the matrix obtained by appending the columns of b to the right of those of A.

So for our system the augmented matrix is:

To transform the augmented matrix to reduced row echelon form we need to follow this row operations:

add -1 times the 1st row to the 2nd row

add -2 times the 1st row to the 3rd row

multiply the 2nd row by -1/2

add 2 times the 2nd row to the 3rd row

multiply the 3rd row by 1/2

add -3/2 times the 3rd row to the 2nd row

add -1 times the 3rd row to the 1st row

add -1 times the 2nd row to the 1st row

Find the solutions set and put in vector form.

Interpret the reduced row echelon form:

The reduced row echelon form of the augmented matrix is

which corresponds to the system:

We can solve for z:

and replace this value into the other two equations

No equation of this system has a form zero = nonzero; Therefore, the system is consistent. The system has infinitely many solutions:

where u and w are free variables.

We put all 5 variables into a column vector, in order, x,y,w,z,u

Next we break it up into 3 vectors, the one with all u's, the one with all w's and the one with all constants:

Next we factor u out of the first vector and w out of the second:

The vector form of the general solution is

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