Answer:
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:
- <em>Convert the given system of equations to matrix form</em>
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
- <em>Use row operations to put the augmented matrix in echelon form.</em>
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
- <em>Find the solutions set and put in vector form.</em>
<u>Interpret the reduced row echelon form:</u>
The reduced row echelon form of the augmented matrix is
which corresponds to the system:
We can solve for <em>z:</em>
<em></em>
and replace this value into the other two equations
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No equation of this system has a form zero = nonzero; Therefore, the system is consistent. The system has infinitely many solutions:
<em></em>
where <em>u</em> and <em>w</em> 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 <em>u</em> out of the first vector and <em>w</em> out of the second:
The vector form of the general solution is