Question

Solve the augmented matrix below
6 -12 24
-1 -6 4

Answer 1

To solve the augmented matrix, we start with the matrix:

\[
\begin{bmatrix}
6 & -12 & | & 24 \\
-1 & -6 & | & 4
\end{bmatrix}
\]

This corresponds to the system of equations:

1. \( 6x - 12y = 24 \)
2. \(-x - 6y = 4 \)

We can use row operations to simplify this augmented matrix. Let's start by making the first entry in the first row equal to 1 by dividing the first row by 6:

\[
\begin{bmatrix}
1 & -2 & | & 4 \\
-1 & -6 & | & 4
\end{bmatrix}
\]

Next, we can eliminate the \(-1\) in the second row, first column by adding the first row to the second row:

\[
\begin{bmatrix}
1 & -2 & | & 4 \\
0 & -8 & | & 8
\end{bmatrix}
\]

Now, we can simplify the second row by dividing it by \(-8\):

\[
\begin{bmatrix}
1 & -2 & | & 4 \\
0 & 1 & | & -1
\end{bmatrix}
\]

Now we can back substitute. The second equation \( y = -1 \). We can substitute \( y \) into the first equation:

\[
x - 2(-1) = 4
\]
\[
x + 2 = 4
\]
\[
x = 2
\]

Therefore, the solutions to the system of equations are:

\[
x = 2, \quad y = -1
\]

In coordinate form, the solution is \( (2, -1) \).