Answer :
To solve this problem, let's break down the given conditions and interpret the reduced row echelon form of the matrix which gives us the solution.
1. Understanding the Problem:
- The store sells almonds at \[tex]$7 per pound, cashews at \$[/tex]10 per pound, and walnuts at \[tex]$12 per pound.
- The customer buys 12 pounds of mixed nuts and spends \$[/tex]118.
- The customer buys 2 more pounds of walnuts than cashews.
2. Setting the Equations:
- Let [tex]\( A \)[/tex] be the pounds of almonds, [tex]\( C \)[/tex] be the pounds of cashews, and [tex]\( W \)[/tex] be the pounds of walnuts.
- From the problem, we have the following equations:
1. Nuts weight equation: [tex]\( A + C + W = 12 \)[/tex]
2. Cost equation: [tex]\( 7A + 10C + 12W = 118 \)[/tex]
3. Relationship between walnuts and cashews: [tex]\( W = C + 2 \)[/tex]
3. Interpreting the Reduced Row Echelon Form Matrix:
- The matrix represents the system of equations in a form that is ready to be interpreted directly:
```
⎡ 0 -1 1 | 2 ⎤
⎢ 7 10 12 | 118 ⎥
⎣ 1 1 1 | 12 ⎦
```
- The first row (0, -1, 1, | 2) corresponds to the condition [tex]\( W = C + 2 \)[/tex].
- The other rows help solve for the values of A, C, and W.
4. Solving for the Number of Pounds:
- From the row [tex]\( W = C + 2 \)[/tex], if [tex]\( C = 3 \)[/tex] then:
- [tex]\( W = 3 + 2 = 5 \)[/tex]
- Using the equation [tex]\( A + C + W = 12 \)[/tex]:
- [tex]\( A + 3 + 5 = 12 \)[/tex]
- [tex]\( A = 12 - 8 = 4 \)[/tex]
5. Result Analysis:
- The customer buys:
- 4 pounds of almonds,
- 3 pounds of cashews,
- 5 pounds of walnuts.
6. Interpretation of Results:
- The difference between the pounds of walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].
- The difference between the pounds of almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].
Thus, the statement describes the situation accurately: "The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
1. Understanding the Problem:
- The store sells almonds at \[tex]$7 per pound, cashews at \$[/tex]10 per pound, and walnuts at \[tex]$12 per pound.
- The customer buys 12 pounds of mixed nuts and spends \$[/tex]118.
- The customer buys 2 more pounds of walnuts than cashews.
2. Setting the Equations:
- Let [tex]\( A \)[/tex] be the pounds of almonds, [tex]\( C \)[/tex] be the pounds of cashews, and [tex]\( W \)[/tex] be the pounds of walnuts.
- From the problem, we have the following equations:
1. Nuts weight equation: [tex]\( A + C + W = 12 \)[/tex]
2. Cost equation: [tex]\( 7A + 10C + 12W = 118 \)[/tex]
3. Relationship between walnuts and cashews: [tex]\( W = C + 2 \)[/tex]
3. Interpreting the Reduced Row Echelon Form Matrix:
- The matrix represents the system of equations in a form that is ready to be interpreted directly:
```
⎡ 0 -1 1 | 2 ⎤
⎢ 7 10 12 | 118 ⎥
⎣ 1 1 1 | 12 ⎦
```
- The first row (0, -1, 1, | 2) corresponds to the condition [tex]\( W = C + 2 \)[/tex].
- The other rows help solve for the values of A, C, and W.
4. Solving for the Number of Pounds:
- From the row [tex]\( W = C + 2 \)[/tex], if [tex]\( C = 3 \)[/tex] then:
- [tex]\( W = 3 + 2 = 5 \)[/tex]
- Using the equation [tex]\( A + C + W = 12 \)[/tex]:
- [tex]\( A + 3 + 5 = 12 \)[/tex]
- [tex]\( A = 12 - 8 = 4 \)[/tex]
5. Result Analysis:
- The customer buys:
- 4 pounds of almonds,
- 3 pounds of cashews,
- 5 pounds of walnuts.
6. Interpretation of Results:
- The difference between the pounds of walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].
- The difference between the pounds of almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].
Thus, the statement describes the situation accurately: "The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
