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]
\[
\begin{bmatrix}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{bmatrix}
\]
[/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?

To solve the problem, we need to find out how many pounds of almonds, cashews, and walnuts the customer buys, given the information in the matrix.

The matrix represents the following system of equations:

1. [tex]$0x - 1y + 1z = 2$[/tex] (Equation 1)
2. [tex]$7x + 10y + 12z = 118$[/tex] (Equation 2)
3. [tex]$x + y + z = 12$[/tex] (Equation 3)

Let's interpret these equations:

- Equation 1 represents the condition for walnuts being 2 more pounds than cashews: [tex]$-y + z = 2$[/tex], which means [tex]$z = y + 2$[/tex].
- Equation 3 represents the total weight of nuts bought: [tex]$x + y + z = 12$[/tex].
- Equation 2 represents the total cost of the nuts: [tex]$7x + 10y + 12z = 118$[/tex].

Now, we'll solve the system step by step:

Step 1: Solve Equation 1 for [tex]$z$[/tex]:

[tex]\[ z = y + 2 \][/tex]

Step 2: Substitute [tex]$z = y + 2$[/tex] into Equations 2 and 3.

Substitute into Equation 3:

[tex]\[ x + y + (y + 2) = 12 \][/tex]

Simplify:

[tex]\[ x + 2y + 2 = 12 \][/tex]

[tex]\[ x + 2y = 10 \][/tex] (Equation 3a)

Substitute into Equation 2:

[tex]\[ 7x + 10y + 12(y + 2) = 118 \][/tex]

Expand and simplify:

[tex]\[ 7x + 10y + 12y + 24 = 118 \][/tex]

[tex]\[ 7x + 22y + 24 = 118 \][/tex]

Subtract 24 from both sides:

[tex]\[ 7x + 22y = 94 \][/tex] (Equation 2a)

Step 3: Now solve the system of the two equations we just found:

[tex]\[
\left\{
\begin{array}{l}
x + 2y = 10 \\
7x + 22y = 94 \\
\end{array}
\right.
\][/tex]

Use Equation 3a to express [tex]$x$[/tex] in terms of [tex]$y$[/tex]:

[tex]\[ x = 10 - 2y \][/tex]

Substitute [tex]$x = 10 - 2y$[/tex] into Equation 2a:

[tex]\[ 7(10 - 2y) + 22y = 94 \][/tex]

[tex]\[ 70 - 14y + 22y = 94 \][/tex]

Combine like terms:

[tex]\[ 70 + 8y = 94 \][/tex]

Subtract 70 from both sides:

[tex]\[ 8y = 24 \][/tex]

Divide by 8:

[tex]\[ y = 3 \][/tex]

Step 4: Now find [tex]$x$[/tex] and [tex]$z$[/tex]:

- For [tex]$x$[/tex]:

[tex]\[ x = 10 - 2y = 10 - 2(3) = 10 - 6 = 4 \][/tex]

- For [tex]$z$[/tex]:

[tex]\[ z = y + 2 = 3 + 2 = 5 \][/tex]

So, the customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.