Answer :
To solve this problem, we need to interpret the given matrix and determine how many pounds of each type of nut were bought by the customer. Let's go through the steps one by one:
1. Understand the Information:
- The store sells three types of nuts: almonds, cashews, and walnuts with prices 7, 10, and 12 dollars per pound respectively.
- The total weight of the nuts is 12 pounds, and the total cost is [tex]$118.
- There are 2 more pounds of walnuts than cashews.
2. Set Up the Equations:
- Let's denote the pounds of almonds, cashews, and walnuts as \( a \), \( c \), and \( w \) respectively.
From the problem, we can form the following equations:
- Equation 1: The customer buys 2 more pounds of walnuts than cashews: \( w = c + 2 \).
- Equation 2: The total cost of the nuts is $[/tex]118: [tex]\( 7a + 10c + 12w = 118 \)[/tex].
- Equation 3: The total weight of the nuts is 12 pounds: [tex]\( a + c + w = 12 \)[/tex].
3. Matrix Representation:
The matrix given in the question corresponds to the system of equations:
```
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\ \quad \text{(from [tex]\( w = c + 2 \)[/tex])}
7 & 10 & 12 & 118 \\ \quad \text{(from the cost equation)}
1 & 1 & 1 & 12 \quad \text{(from the total weight)}
\end{array}\right]
```
4. Solution:
According to the results from the operations (without referring to any code or calculations):
- The customer buys 4 pounds of almonds ([tex]\( a = 4 \)[/tex]).
- The customer buys 3 pounds of cashews ([tex]\( c = 3 \)[/tex]).
- The customer buys 5 pounds of walnuts ([tex]\( w = 5 \)[/tex]).
5. Verifying the Statements:
- Let’s compare these values with the statements provided:
- The difference between walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].
- The difference between almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].
Thus, the correct interpretation of the results is: The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
1. Understand the Information:
- The store sells three types of nuts: almonds, cashews, and walnuts with prices 7, 10, and 12 dollars per pound respectively.
- The total weight of the nuts is 12 pounds, and the total cost is [tex]$118.
- There are 2 more pounds of walnuts than cashews.
2. Set Up the Equations:
- Let's denote the pounds of almonds, cashews, and walnuts as \( a \), \( c \), and \( w \) respectively.
From the problem, we can form the following equations:
- Equation 1: The customer buys 2 more pounds of walnuts than cashews: \( w = c + 2 \).
- Equation 2: The total cost of the nuts is $[/tex]118: [tex]\( 7a + 10c + 12w = 118 \)[/tex].
- Equation 3: The total weight of the nuts is 12 pounds: [tex]\( a + c + w = 12 \)[/tex].
3. Matrix Representation:
The matrix given in the question corresponds to the system of equations:
```
\left[\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\ \quad \text{(from [tex]\( w = c + 2 \)[/tex])}
7 & 10 & 12 & 118 \\ \quad \text{(from the cost equation)}
1 & 1 & 1 & 12 \quad \text{(from the total weight)}
\end{array}\right]
```
4. Solution:
According to the results from the operations (without referring to any code or calculations):
- The customer buys 4 pounds of almonds ([tex]\( a = 4 \)[/tex]).
- The customer buys 3 pounds of cashews ([tex]\( c = 3 \)[/tex]).
- The customer buys 5 pounds of walnuts ([tex]\( w = 5 \)[/tex]).
5. Verifying the Statements:
- Let’s compare these values with the statements provided:
- The difference between walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].
- The difference between almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].
Thus, the correct interpretation of the results is: The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
