Question

write an augmented matrix and use elementary row operations in order to solve the following system of equations. your final matrix should be in reduced row echelon form. in order to get credit you will have to have a correct final answer as accurate steps in each row operation.
$x-3y+z = -5$
$x-4y = 0$
$x-5y = -1$
write the augmented matrix:

Explanation:

Step1: Form the augmented matrix

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 1 & -4 & 0 & \mid & 0 \\ 1 & -5 & 0 & \mid & -1 \end{bmatrix}$$

Step2: Subtract Row1 from Row2

$R_2 = R_2 - R_1$

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 0 & -1 & -1 & \mid & 5 \\ 1 & -5 & 0 & \mid & -1 \end{bmatrix}$$

Step3: Subtract Row1 from Row3

$R_3 = R_3 - R_1$

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 0 & -1 & -1 & \mid & 5 \\ 0 & -2 & -1 & \mid & 4 \end{bmatrix}$$

Step4: Multiply Row2 by -1

$R_2 = -R_2$

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 0 & 1 & 1 & \mid & -5 \\ 0 & -2 & -1 & \mid & 4 \end{bmatrix}$$

Step5: Add 2×Row2 to Row3

$R_3 = R_3 + 2R_2$

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 0 & 1 & 1 & \mid & -5 \\ 0 & 0 & 1 & \mid & -6 \end{bmatrix}$$

Step6: Subtract Row3 from Row2

$R_2 = R_2 - R_3$

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 0 & 1 & 0 & \mid & 1 \\ 0 & 0 & 1 & \mid & -6 \end{bmatrix}$$

Step7: Subtract Row3, add 3×Row2 to Row1

$R_1 = R_1 - R_3 + 3R_2$

$$\begin{bmatrix} 1 & 0 & 0 & \mid & 4 \\ 0 & 1 & 0 & \mid & 1 \\ 0 & 0 & 1 & \mid & -6 \end{bmatrix}$$

Answer:

The augmented matrix (initial) is:

$$\begin{bmatrix} 1 & -3 & 1 & \mid & -5 \\ 1 & -4 & 0 & \mid & 0 \\ 1 & -5 & 0 & \mid & -1 \end{bmatrix}$$

The reduced row echelon form is:

$$\begin{bmatrix} 1 & 0 & 0 & \mid & 4 \\ 0 & 1 & 0 & \mid & 1 \\ 0 & 0 & 1 & \mid & -6 \end{bmatrix}$$

The solution to the system is $x=4$, $y=1$, $z=-6$