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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 :

Let the number of pounds of almonds, cashews, and walnuts be [tex]$a$[/tex], [tex]$c$[/tex], and [tex]$w$[/tex], respectively. We are given the following three equations:

[tex]$$
a + c + w = 12 \quad \text{(total pounds)}
$$[/tex]

[tex]$$
7a + 10c + 12w = 118 \quad \text{(total cost)}
$$[/tex]

[tex]$$
w - c = 2 \quad \text{(2 more pounds of walnuts than cashews)}
$$[/tex]

Step 1. Express one variable in terms of another.
From the equation

[tex]$$
w-c=2,
$$[/tex]

we get

[tex]$$
w = c + 2.
$$[/tex]

Step 2. Substitute into the total pounds equation.
Replace [tex]$w$[/tex] in

[tex]$$
a+c+w=12
$$[/tex]

by [tex]$c+2$[/tex] to obtain:

[tex]$$
a + c + (c + 2) = 12 \quad \implies \quad a + 2c + 2 = 12.
$$[/tex]

Subtracting 2 from both sides gives:

[tex]$$
a + 2c = 10 \quad \implies \quad a = 10 - 2c.
$$[/tex]

Step 3. Substitute [tex]$a$[/tex] and [tex]$w$[/tex] into the total cost equation.
Substitute [tex]$a = 10 - 2c$[/tex] and [tex]$w = c+2$[/tex] into

[tex]$$
7a + 10c + 12w = 118:
$$[/tex]

[tex]$$
7(10 - 2c) + 10c + 12(c+2) = 118.
$$[/tex]

Expanding and simplifying:

[tex]$$
70 - 14c + 10c + 12c + 24 = 118.
$$[/tex]

Combine like terms:

[tex]$$
94 + 8c = 118.
$$[/tex]

Subtract 94 from both sides:

[tex]$$
8c = 24 \quad \implies \quad c = 3.
$$[/tex]

Step 4. Solve for [tex]$a$[/tex] and [tex]$w$[/tex].
Now, substitute [tex]$c = 3$[/tex] back into the expressions for [tex]$a$[/tex] and [tex]$w$[/tex]:

[tex]$$
a = 10 - 2(3) = 4,
$$[/tex]

[tex]$$
w = 3 + 2 = 5.
$$[/tex]

Step 5. Interpret the differences.
We now compare the weights:

- The difference between walnuts and almonds is

[tex]$$
w - a = 5 - 4 = 1.
$$[/tex]

- The difference between almonds and cashews is

[tex]$$
a - c = 4 - 3 = 1.
$$[/tex]

Thus, the customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

Final Answer:
The interpretation is:
"The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."

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