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A store sells almonds for [tex]\$7[/tex] per pound, cashews for [tex]\$10[/tex] per pound, and walnuts for [tex]\$12[/tex] per pound. A customer buys 12 pounds of mixed nuts consisting of almonds, cashews, and walnuts for [tex]\$118[/tex]. The customer buys 2 more pounds of walnuts than cashews. The matrix below represents this situation.

[tex]
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}\right]
[/tex]

If the reduced row echelon form of this matrix represents the amount of each type of nut the customer buys, which statement is a possible interpretation of the results?

A. The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

B. The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.

C. The customer buys 0.5 more pound of walnuts than almonds and 2.5 more pounds of almonds than cashews.

D. The customer buys 6.5 more pounds of walnuts than almonds and 8.5 more pounds of almonds than cashews.

Answer :

To solve this problem, we'll translate the given information into a set of equations and then analyze the provided augmented matrix:

### Step-by-Step Solution:

1. Formulate the Equations:

Let's denote:
- [tex]\( A \)[/tex] = pounds of almonds
- [tex]\( C \)[/tex] = pounds of cashews
- [tex]\( W \)[/tex] = pounds of walnuts

From the given problem, we have the following information:

- The total amounts to 12 pounds:
[tex]\[
A + C + W = 12
\][/tex]

- The total cost is \$118:
[tex]\[
7A + 10C + 12W = 118
\][/tex]

- The customer buys 2 more pounds of walnuts than cashews:
[tex]\[
W = C + 2
\][/tex]

2. Set Up the Augmented Matrix:

These equations are represented by the following augmented matrix:
[tex]\[
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}\right]
\][/tex]

3. Solve the System using RREF:

The reduced row echelon form (RREF) of this matrix will help us determine the values of [tex]\( A \)[/tex], [tex]\( C \)[/tex], and [tex]\( W \)[/tex]. Using analytical methods or computational tools, we derive the RREF matrix.

4. Extract Variable Values:

After deriving the RREF for this matrix, the values of [tex]\( A \)[/tex], [tex]\( C \)[/tex], and [tex]\( W \)[/tex] can be read directly.

5. Analyze the Results:

With the given answers, after interpreting the results, we find the following possible statements:

- The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
- The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.
- The customer buys 0.5 more pound of walnuts than almonds and 2.5 more pounds of almonds than cashews.
- The customer buys 6.5 more pounds of walnuts than almonds and 8.5 more pounds of almonds than cashews.

Given the calculated results, all these interpretations are found to be false:

Therefore, the correct interpretation would be none of the listed statements apply:

The provided conditions and constraints don't match any of these results.

In summary, based on the provided calculations, none of the statements accurately describe the distribution of the almond, cashew, and walnut amounts purchased by the customer.

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