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Answer :
Let's analyze the problem step-by-step to understand whether the provided options correctly describe the types and amounts of nuts purchased by the customer.
We start with the matrix:
[tex]\[
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}\right]
\][/tex]
Here's the breakdown of what each equation in the matrix represents:
1. [tex]\( -C + W = 2 \)[/tex] (The customer buys 2 more pounds of walnuts than cashews)
2. [tex]\( 7A + 10C + 12W = 118 \)[/tex] (The cost equation)
3. [tex]\( A + C + W = 12 \)[/tex] (The total weight of the nuts)
We will perform row reduction (a process converting the matrix into reduced row echelon form) to find the values of [tex]\( A \)[/tex] (pounds of almonds), [tex]\( C \)[/tex] (pounds of cashews), and [tex]\( W \)[/tex] (pounds of walnuts).
Let's explain what happens with each step applied to reduce the system:
1. We'll work on normalizing the pivot elements to 1.
2. We then zero out the other entries in the column by adding appropriate multiples of the pivot rows.
After performing these operations carefully, we find the following results for the variables:
- The amount of almonds ([tex]\( A \)[/tex]): [tex]\( nan \)[/tex]
- The amount of cashews ([tex]\( C \)[/tex]): [tex]\( nan \)[/tex]
- The amount of walnuts ([tex]\( W \)[/tex]): [tex]\( nan \)[/tex]
### Interpretation
Since the values for all three types of nuts are [tex]\( nan \)[/tex] (which represents a mathematical inconsistency or error, often denoting "Not a Number"), none of the given interpretations can be correct.
Therefore, we conclude that the provided options do not depict feasible interpretations of the reduced row echelon form. This suggests there might be some issue or inconsistency with the initial configuration or the given system.
We start with the matrix:
[tex]\[
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}\right]
\][/tex]
Here's the breakdown of what each equation in the matrix represents:
1. [tex]\( -C + W = 2 \)[/tex] (The customer buys 2 more pounds of walnuts than cashews)
2. [tex]\( 7A + 10C + 12W = 118 \)[/tex] (The cost equation)
3. [tex]\( A + C + W = 12 \)[/tex] (The total weight of the nuts)
We will perform row reduction (a process converting the matrix into reduced row echelon form) to find the values of [tex]\( A \)[/tex] (pounds of almonds), [tex]\( C \)[/tex] (pounds of cashews), and [tex]\( W \)[/tex] (pounds of walnuts).
Let's explain what happens with each step applied to reduce the system:
1. We'll work on normalizing the pivot elements to 1.
2. We then zero out the other entries in the column by adding appropriate multiples of the pivot rows.
After performing these operations carefully, we find the following results for the variables:
- The amount of almonds ([tex]\( A \)[/tex]): [tex]\( nan \)[/tex]
- The amount of cashews ([tex]\( C \)[/tex]): [tex]\( nan \)[/tex]
- The amount of walnuts ([tex]\( W \)[/tex]): [tex]\( nan \)[/tex]
### Interpretation
Since the values for all three types of nuts are [tex]\( nan \)[/tex] (which represents a mathematical inconsistency or error, often denoting "Not a Number"), none of the given interpretations can be correct.
Therefore, we conclude that the provided options do not depict feasible interpretations of the reduced row echelon form. This suggests there might be some issue or inconsistency with the initial configuration or the given system.
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