A store sells almonds for [tex]$\$ 7$[/tex] per pound, cashews for [tex]$\[tex]$ 10$[/tex][/tex] per pound, and walnuts for [tex]$\$ 12$[/tex] per pound. A customer buys 12 pounds of mixed nuts consisting of almonds, cashews, and walnuts for [tex]$\[tex]$ 118$[/tex][/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.

To solve the problem, we need to determine how many pounds of each type of nut the customer buys. Let's break it down step-by-step based on the information given:

1. Define the Variables:

- Let [tex]$ A $[/tex] represent the pounds of almonds.
- Let [tex]$ C $[/tex] represent the pounds of cashews.
- Let [tex]$ W $[/tex] represent the pounds of walnuts.

2. Set Up the Equations:

From the problem, we have the following conditions:

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

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

- The total weight of the nuts bought is 12 pounds:
[tex]\[
A + C + W = 12
\][/tex]

3. Substitute and Solve:

We need to use the equations to find the values of [tex]$ A $[/tex], [tex]$ C $[/tex], and [tex]$ W $[/tex]:

- Substitute [tex]$ W = C + 2 $[/tex] into the second and third equations:
[tex]\[
7A + 10C + 12(C + 2) = 118
\][/tex]
[tex]\[
A + C + (C + 2) = 12
\][/tex]

- Simplify these equations:
- For the cost equation:
[tex]\[
7A + 10C + 12C + 24 = 118
\][/tex]
[tex]\[
7A + 22C = 94
\][/tex]

- For the weight equation:
[tex]\[
A + 2C + 2 = 12
\][/tex]
[tex]\[
A + 2C = 10
\][/tex]

4. Solve for [tex]$ A $[/tex] and [tex]$ C $[/tex]:

- Use elimination or substitution to solve the equations [tex]$ 7A + 22C = 94 $[/tex] and [tex]$ A + 2C = 10 $[/tex].

- From [tex]$ A + 2C = 10 $[/tex], we can express [tex]$ A $[/tex] as:
[tex]\[
A = 10 - 2C
\][/tex]

- Substitute [tex]$ A = 10 - 2C $[/tex] into [tex]$ 7A + 22C = 94 $[/tex]:
[tex]\[
7(10 - 2C) + 22C = 94
\][/tex]
[tex]\[
70 - 14C + 22C = 94
\][/tex]
[tex]\[
70 + 8C = 94
\][/tex]
[tex]\[
8C = 24
\][/tex]
[tex]\[
C = 3
\][/tex]

- Substitute [tex]$ C = 3 $[/tex] back to find [tex]$ A $[/tex]:
[tex]\[
A = 10 - 2(3)
\][/tex]
[tex]\[
A = 4
\][/tex]

5. Find [tex]$ W $[/tex]:

Since [tex]$ W = C + 2 $[/tex]:
[tex]\[
W = 3 + 2 = 5
\][/tex]

6. Interpret the Solution:

- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- This means the customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

Therefore, the correct interpretation is: "The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."