Question

solve the system of equations by finding the reduced row-echelon form of the augmented matrix for the system of equations.
$3x - 4y - 5z = -27$
$5x + 2y - 2z = 11$
$5x - 4y + 4z = -7$
a. $(1, 5, 51)$ \t\t\t\t c. $(10, 51, 23)$
b. $(10, 5, 51)$ \t\t\t\t d. $(1, 5, 2)$

please select the best answer from the choices provided

a
b
c

Explanation:

Step1: Write augmented matrix

$$\begin{bmatrix} 3 & -4 & -5 & \mid & -27 \\ 5 & 2 & -2 & \mid & 11 \\ 5 & -4 & 4 & \mid & -7 \end{bmatrix}$$

Step2: Normalize row1 (divide by 3)

$$\begin{bmatrix} 1 & -\frac{4}{3} & -\frac{5}{3} & \mid & -9 \\ 5 & 2 & -2 & \mid & 11 \\ 5 & -4 & 4 & \mid & -7 \end{bmatrix}$$

Step3: Eliminate $x$ from row2/3

Row2 = Row2 - 5Row1, Row3 = Row3 - 5Row1

$$\begin{bmatrix} 1 & -\frac{4}{3} & -\frac{5}{3} & \mid & -9 \\ 0 & \frac{26}{3} & \frac{19}{3} & \mid & 56 \\ 0 & \frac{8}{3} & \frac{37}{3} & \mid & 38 \end{bmatrix}$$

Step4: Normalize row2 (multiply by $\frac{3}{26}$)

$$\begin{bmatrix} 1 & -\frac{4}{3} & -\frac{5}{3} & \mid & -9 \\ 0 & 1 & \frac{19}{26} & \mid & \frac{84}{13} \\ 0 & \frac{8}{3} & \frac{37}{3} & \mid & 38 \end{bmatrix}$$

Step5: Eliminate $y$ from row1/3

Row1 = Row1 + $\frac{4}{3}$Row2, Row3 = Row3 - $\frac{8}{3}$Row2

$$\begin{bmatrix} 1 & 0 & -\frac{17}{13} & \mid & -\frac{25}{13} \\ 0 & 1 & \frac{19}{26} & \mid & \frac{84}{13} \\ 0 & 0 & \frac{195}{26} & \mid & \frac{195}{13} \end{bmatrix}$$

Step6: Normalize row3 (multiply by $\frac{26}{195}$)

$$\begin{bmatrix} 1 & 0 & -\frac{17}{13} & \mid & -\frac{25}{13} \\ 0 & 1 & \frac{19}{26} & \mid & \frac{84}{13} \\ 0 & 0 & 1 & \mid & 2 \end{bmatrix}$$

Step7: Eliminate $z$ from row1/2

Row1 = Row1 + $\frac{17}{13}$Row3, Row2 = Row2 - $\frac{19}{26}$Row3

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

Answer:

d. (1, 5, 2)