### Part A: Express the Equations as an Augmented Matrix
To express the given system of equations as an augmented matrix, we need to arrange the coefficients of the variables (x and y) and the constants in matrix form.
The given system of equations is:
[tex]\[ 2x - 4y = 7 \][/tex]
[tex]\[ -3x + y = -1 \][/tex]
From the first equation [tex]\(2x - 4y = 7\)[/tex]:
- The coefficient of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
- The coefficient of [tex]\(y\)[/tex] is [tex]\(-4\)[/tex].
- The constant term is [tex]\(7\)[/tex].
From the second equation [tex]\(-3x + y = -1\)[/tex]:
- The coefficient of [tex]\(x\)[/tex] is [tex]\(-3\)[/tex].
- The coefficient of [tex]\(y\)[/tex] is [tex]\(1\)[/tex].
- The constant term is [tex]\(-1\)[/tex].
We organize these coefficients and constants into an augmented matrix:
[tex]\[
\begin{bmatrix}
2 & -4 & \vert & 7 \\
-3 & 1 & \vert & -1
\end{bmatrix}
\][/tex]
In simplified notation, we omit the separator line for notation purposes:
[tex]\[
\left[\begin{array}{ccc}
2 & -4 & 7 \\
-3 & 1 & -1
\end{array}\right]
\][/tex]
### Part B: Multiply the Augmented Matrix by the Scalar 3
Next, we need to multiply the augmented matrix by a scalar, which is given as [tex]\(3\)[/tex].
The original augmented matrix is:
[tex]\[
\begin{bmatrix}
2 & -4 & 7 \\
-3 & 1 & -1
\end{bmatrix}
\][/tex]
We will multiply each entry in this matrix by the scalar [tex]\(3\)[/tex].
First row:
- [tex]\(2 \times 3 = 6\)[/tex]
- [tex]\(-4 \times 3 = -12\)[/tex]
- [tex]\(7 \times 3 = 21\)[/tex]
Second row:
- [tex]\(-3 \times 3 = -9\)[/tex]
- [tex]\(1 \times 3 = 3\)[/tex]
- [tex]\(-1 \times 3 = -3\)[/tex]
So, the scaled augmented matrix is:
[tex]\[
\begin{bmatrix}
6 & -12 & 21 \\
-9 & 3 & -3
\end{bmatrix}
\][/tex]
### Summary of Results
1. The augmented matrix representing the given system of equations is:
[tex]\[
\left[\begin{array}{ccc}
2 & -4 & 7 \\
-3 & 1 & -1
\end{array}\right]
\][/tex]
2. When this matrix is multiplied by the scalar [tex]\(3\)[/tex], the resulting matrix is:
[tex]\[
\left[\begin{array}{ccc}
6 & -12 & 21 \\
-9 & 3 & -3
\end{array}\right]
\][/tex]
