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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 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 2 more pounds of walnuts than cashews.

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

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

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

Answer :

To solve this problem, let's break down the given conditions and interpret the reduced row echelon form of the matrix which gives us the solution.

1. Understanding the Problem:
- The store sells almonds at \[tex]$7 per pound, cashews at \$[/tex]10 per pound, and walnuts at \[tex]$12 per pound.
- The customer buys 12 pounds of mixed nuts and spends \$[/tex]118.
- The customer buys 2 more pounds of walnuts than cashews.

2. Setting the Equations:
- Let [tex]\( A \)[/tex] be the pounds of almonds, [tex]\( C \)[/tex] be the pounds of cashews, and [tex]\( W \)[/tex] be the pounds of walnuts.
- From the problem, we have the following equations:
1. Nuts weight equation: [tex]\( A + C + W = 12 \)[/tex]
2. Cost equation: [tex]\( 7A + 10C + 12W = 118 \)[/tex]
3. Relationship between walnuts and cashews: [tex]\( W = C + 2 \)[/tex]

3. Interpreting the Reduced Row Echelon Form Matrix:
- The matrix represents the system of equations in a form that is ready to be interpreted directly:
```
⎡ 0 -1 1 | 2 ⎤
⎢ 7 10 12 | 118 ⎥
⎣ 1 1 1 | 12 ⎦
```
- The first row (0, -1, 1, | 2) corresponds to the condition [tex]\( W = C + 2 \)[/tex].
- The other rows help solve for the values of A, C, and W.

4. Solving for the Number of Pounds:
- From the row [tex]\( W = C + 2 \)[/tex], if [tex]\( C = 3 \)[/tex] then:
- [tex]\( W = 3 + 2 = 5 \)[/tex]
- Using the equation [tex]\( A + C + W = 12 \)[/tex]:
- [tex]\( A + 3 + 5 = 12 \)[/tex]
- [tex]\( A = 12 - 8 = 4 \)[/tex]

5. Result Analysis:
- The customer buys:
- 4 pounds of almonds,
- 3 pounds of cashews,
- 5 pounds of walnuts.

6. Interpretation of Results:
- The difference between the pounds of walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].
- The difference between the pounds of almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].

Thus, the statement describes the situation accurately: "The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."

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