To determine which system of equations is represented by the given augmented matrix, we need to convert the matrix into its corresponding system of linear equations.
The given augmented matrix is:
[tex]\[
\left[\begin{array}{rr|r}
1 & -3 & 10 \\
2 & 2 & -4
\end{array}\right]
\][/tex]
This matrix contains two rows, each representing a linear equation. The columns of the matrix correspond to the coefficients of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and the constants (right-hand side of the equation) respectively.
Let's break down each row of the matrix to form the equations:
1. For the first row: [tex]\(1 \, x + (-3) \, y = 10\)[/tex]
This simplifies to:
[tex]\[
x - 3y = 10
\][/tex]
2. For the second row: [tex]\(2 \, x + 2 \, y = -4\)[/tex]
This simplifies to:
[tex]\[
2x + 2y = -4
\][/tex]
Thus, the system of linear equations represented by the matrix is:
[tex]\[
\begin{cases}
x - 3y = 10 \\
2x + 2y = -4
\end{cases}
\][/tex]
Now, let's compare this system to the given options:
A. [tex]\(x - 3y = 10\)[/tex] and [tex]\(2x + 2y = -4\)[/tex]
B. [tex]\(1 - 3y = 10\)[/tex] and [tex]\(2x + 2 = -4\)[/tex]
C. [tex]\(y - 3x = 10\)[/tex] and [tex]\(2y + 2x = -4\)[/tex]
As we see, option A perfectly matches the system of equations we derived from the matrix.
Therefore, the correct system of equations represented by the matrix is:
[tex]\[
\boxed{A. \, x - 3y = 10 \, \, \text{and} \, 2x + 2y = -4}
\][/tex]
