Question

Solve the system by finding the reduced row-echelon form of the augmented matrix.

Answer 1

Answer:

The reduced row-echelon form is

$\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right]$

The solutions to the system of equations are:

$y=-2z-3,\:x=-6z-10$

Step-by-step explanation:

To solve this system of linear equations,

$x-4y-2z=2\\2x-11y-10z=13\\-x+6y+6z=-8$

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.

$\left[ \begin{array}{cccc} 1 & -4 & -2 & 2 \\\\ 2 & -11 & -10 & 13 \\\\ -1 & 6 & 6 & -8 \end{array} \right]$

It can be transformed by a sequence of elementary row operations to the matrix.

There are three kinds of elementary matrix operations.

1. Interchange two rows (or columns).
2. Multiply each element in a row (or column) by a non-zero number.
3. 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

$\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right]$

Step 2: Interpret the reduced row echelon form

The reduced row echelon form of the augmented matrix is

$\left[ \begin{array}{cccc} 1 & 0 & 6 & -10 \\\\ 0 & 1 & 2 & -3 \\\\ 0 & 0 & 0 & 0 \end{array} \right]$

which corresponds to the system

$x+6z=-10\\y+2z=-3\\0=0$

The system has infinitely many solutions.

$y=-2z-3,\:x=-6z-10$