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

Answer :

To solve the problem and find the number of pounds of each type of nut the customer buys, we will work systematically using the matrix provided and the conditions given:

### Step 1: Understand the Equations

From the question, we have the following equations based on the conditions:

1. Equation for the number of pounds:
[tex]\[
a + c + w = 12
\][/tex]
where [tex]\(a\)[/tex], [tex]\(c\)[/tex], and [tex]\(w\)[/tex] represent the pounds of almonds, cashews, and walnuts, respectively.

2. Equation for the cost:
[tex]\[
7a + 10c + 12w = 118
\][/tex]

3. Condition for walnuts being 2 pounds more than cashews:
[tex]\[
w = c + 2
\][/tex]

### Step 2: Substitute and Simplify

Using equation 3, substitute [tex]\(w = c + 2\)[/tex] into the other two equations:

- Substitute into the pounds equation:
[tex]\[
a + c + (c + 2) = 12
\][/tex]
Simplifies to:
[tex]\[
a + 2c + 2 = 12
\][/tex]
[tex]\[
a + 2c = 10 \quad \text{(Equation 4)}
\][/tex]

- Substitute into the cost equation:
[tex]\[
7a + 10c + 12(c + 2) = 118
\][/tex]
Which simplifies to:
[tex]\[
7a + 10c + 12c + 24 = 118
\][/tex]
[tex]\[
7a + 22c = 94 \quad \text{(Equation 5)}
\][/tex]

### Step 3: Solve the System of Equations

We now have a system of two equations:

1. [tex]\(a + 2c = 10\)[/tex]
2. [tex]\(7a + 22c = 94\)[/tex]

Solve for [tex]\(a\)[/tex] in terms of [tex]\(c\)[/tex] using equation 4:
[tex]\[
a = 10 - 2c
\][/tex]

Substitute [tex]\(a = 10 - 2c\)[/tex] into equation 5:
[tex]\[
7(10 - 2c) + 22c = 94
\][/tex]

Simplify and solve for [tex]\(c\)[/tex]:
[tex]\[
70 - 14c + 22c = 94
\][/tex]
[tex]\[
8c = 24
\][/tex]
[tex]\[
c = 3
\][/tex]

Find [tex]\(a\)[/tex] using [tex]\(c = 3\)[/tex]:
[tex]\[
a = 10 - 2(3) = 4
\][/tex]

Find [tex]\(w\)[/tex] using [tex]\(w = c + 2\)[/tex]:
[tex]\[
w = 3 + 2 = 5
\][/tex]

### Conclusion

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

Therefore, this is the interpretation of the results from the given matrix.

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