Certainly! Let's analyze the given linear system of equations and identify the corresponding elements in the coefficient and augmented matrices.
Given system of equations:
[tex]\[
\begin{array}{l}
3x + 7y = 20 \\
x - 4y = 9
\end{array}
\][/tex]
### Coefficient Matrix [tex]\( A \)[/tex]
First, we need to identify the values of the coefficient matrix elements. Each element [tex]\(a_{ij}\)[/tex] represents the coefficient of the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equations.
[tex]\[
A = \begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\][/tex]
From the first equation [tex]\( 3x + 7y = 20 \)[/tex]:
- [tex]\(a_{11} = 3\)[/tex]
- [tex]\(a_{12} = 7\)[/tex]
From the second equation [tex]\( x - 4y = 9 \)[/tex]:
- [tex]\(a_{21} = 1\)[/tex]
- [tex]\(a_{22} = -4\)[/tex]
Thus, the coefficient matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{bmatrix}
3 & 7 \\
1 & -4
\end{bmatrix}
\][/tex]
### Augmented Matrix
Next, we want to identify the values of the augmented matrix elements. The augmented matrix includes the coefficients and the constants from the right-hand side of the equations.
The augmented matrix is:
[tex]\[
\left[ \begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{array} \right]
\][/tex]
From the first equation [tex]\( 3x + 7y = 20 \)[/tex]:
- [tex]\(a_{11} = 3\)[/tex]
- [tex]\(a_{12} = 7\)[/tex]
- [tex]\(a_{13} = 20\)[/tex] (the constant on the right-hand side)
From the second equation [tex]\( x - 4y = 9 \)[/tex]:
- [tex]\(a_{21} = 1\)[/tex]
- [tex]\(a_{22} = -4\)[/tex]
- [tex]\(a_{23} = 9\)[/tex] (the constant on the right-hand side)
Thus, the augmented matrix is:
[tex]\[
\left[ \begin{array}{ccc}
3 & 7 & 20 \\
1 & -4 & 9
\end{array} \right]
\][/tex]
### Summary
Let's summarize the computed values:
Coefficient Matrix:
[tex]\[
A = \begin{bmatrix}
3 & 7 \\
1 & -4
\end{bmatrix}
\][/tex]
Augmented Matrix:
[tex]\[
\left[ \begin{array}{ccc}
3 & 7 & 20 \\
1 & -4 & 9
\end{array} \right]
\][/tex]
Identifying each element:
- [tex]\(a_{11} = 3\)[/tex]
- [tex]\(a_{12} = 7\)[/tex]
- [tex]\(a_{13} = 20\)[/tex]
- [tex]\(a_{21} = 1\)[/tex]
- [tex]\(a_{22} = -4\)[/tex]
- [tex]\(a_{23} = 9\)[/tex]
These values provide a complete representation of the boundaries of the given linear system of equations in terms of its coefficient and augmented matrices.
