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.
$8x+7y = 1$
$-5x-4y = 2$
write the augmented matrix:

Explanation:

Step1: Form the augmented matrix

$$\begin{bmatrix} 8 & 7 & \mid & 1 \\ -5 & -4 & \mid & 2 \end{bmatrix}$$

Step2: Normalize Row1 (divide by 8)

Row1 = $\frac{1}{8}$Row1

$$\begin{bmatrix} 1 & \frac{7}{8} & \mid & \frac{1}{8} \\ -5 & -4 & \mid & 2 \end{bmatrix}$$

Step3: Eliminate $x$ in Row2

Row2 = 5Row1 + Row2

$$\begin{bmatrix} 1 & \frac{7}{8} & \mid & \frac{1}{8} \\ 0 & \frac{3}{8} & \mid & \frac{21}{8} \end{bmatrix}$$

Step4: Normalize Row2 (multiply by $\frac{8}{3}$)

Row2 = $\frac{8}{3}$Row2

$$\begin{bmatrix} 1 & \frac{7}{8} & \mid & \frac{1}{8} \\ 0 & 1 & \mid & 7 \end{bmatrix}$$

Step5: Eliminate $y$ in Row1

Row1 = $-\frac{7}{8}$Row2 + Row1

$$\begin{bmatrix} 1 & 0 & \mid & -6 \\ 0 & 1 & \mid & 7 \end{bmatrix}$$

Answer:

Augmented Matrix (initial):

$$\begin{bmatrix} 8 & 7 & \mid & 1 \\ -5 & -4 & \mid & 2 \end{bmatrix}$$

Reduced Row Echelon Form:

$$\begin{bmatrix} 1 & 0 & \mid & -6 \\ 0 & 1 & \mid & 7 \end{bmatrix}$$

Solution to the system:

$x=-6$, $y=7$