Respuesta :
Answer:
The reduced row-echelon form is
[tex]\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right][/tex]
The solutions to the system of equations are:
[tex]y=-2z-3,\:x=-6z-10[/tex]
Step-by-step explanation:
To solve this system of linear equations,
[tex]x-4y-2z=2\\2x-11y-10z=13\\-x+6y+6z=-8[/tex]
you must:
Step 1: Transform the augmented matrix to the reduced row echelon form.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
This is the augmented matrix that represents the system.
[tex]\left[ \begin{array}{cccc} 1 & -4 & -2 & 2 \\\\ 2 & -11 & -10 & 13 \\\\ -1 & 6 & 6 & -8 \end{array} \right][/tex]
It can be transformed by a sequence of elementary row operations to the matrix.
There are three kinds of elementary matrix operations.
- Interchange two rows (or columns).
- Multiply each element in a row (or column) by a non-zero number.
- Multiply a row (or column) by a non-zero number and add the result to another row (or column).
Using elementary matrix operations, we get that
Row Operation 1: add -2 times the 1st row to the 2nd row
Row Operation 2: add 1 times the 1st row to the 3rd row
Row Operation 3: multiply the 2nd row by -1/3
Row Operation 4: add -2 times the 2nd row to the 3rd row
Row Operation 5: add 4 times the 2nd row to the 1st row
[tex]\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right][/tex]
Step 2: Interpret the reduced row echelon form
The reduced row echelon form of the augmented matrix is
[tex]\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right][/tex]
which corresponds to the system
[tex]x+6z=-10\\y+2z=-3\\0=0[/tex]
The system has infinitely many solutions.
[tex]y=-2z-3,\:x=-6z-10[/tex]
