Answer :
Sure! Let's solve the problem step-by-step using the given matrix to find out how many pounds of each type of nut the customer buys.
### Step 1: Understand the problem
The store sells:
- Almonds for [tex]$7 per pound
- Cashews for $[/tex]10 per pound
- Walnuts for [tex]$12 per pound
The customer buys 12 pounds of mixed nuts (almonds, cashews, and walnuts) for a total of $[/tex]118 and buys 2 more pounds of walnuts than cashews.
### Step 2: Set up the equations
From the problem statement, we can set up the following three equations:
1. [tex]\( x + y + z = 12 \)[/tex] (the total weight is 12 pounds)
2. [tex]\( 7x + 10y + 12z = 118 \)[/tex] (the total cost is $118)
3. [tex]\( z = y + 2 \)[/tex] (2 more pounds of walnuts than cashews)
### Step 3: Represent the equations as a matrix
We form an augmented matrix for these equations:
[tex]\[
\left[
\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}
\right]
\][/tex]
Here, [tex]\( x \)[/tex] represents pounds of almonds, [tex]\( y \)[/tex] represents pounds of cashews, and [tex]\( z \)[/tex] represents pounds of walnuts.
### Step 4: Solve the matrix
To solve for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex], we convert this matrix to reduced row echelon form (RREF). The RREF form will give us the solution directly.
### Step 5: Identify the solution
After converting the matrix to RREF and solving we get:
- [tex]\( x = 4 \)[/tex]
- [tex]\( y = 3 \)[/tex]
- [tex]\( z = 5 \)[/tex]
This means:
- The customer buys 4 pounds of almonds.
- The customer buys 3 pounds of cashews.
- The customer buys 5 pounds of walnuts.
### Step 6: Interpret the results
Now, let's interpret these results to match one of the provided statements:
- The customer buys 5 pounds of walnuts, 4 pounds of almonds, and 3 pounds of cashews.
- Compare walnuts to almonds: 5 - 4 = 1 (walnuts are 1 pound more than almonds)
- Compare almonds to cashews: 4 - 3 = 1 (almonds are 1 pound more than cashews)
So, the correct interpretation is:
- The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
Thus, the correct statement is:
The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
### Step 1: Understand the problem
The store sells:
- Almonds for [tex]$7 per pound
- Cashews for $[/tex]10 per pound
- Walnuts for [tex]$12 per pound
The customer buys 12 pounds of mixed nuts (almonds, cashews, and walnuts) for a total of $[/tex]118 and buys 2 more pounds of walnuts than cashews.
### Step 2: Set up the equations
From the problem statement, we can set up the following three equations:
1. [tex]\( x + y + z = 12 \)[/tex] (the total weight is 12 pounds)
2. [tex]\( 7x + 10y + 12z = 118 \)[/tex] (the total cost is $118)
3. [tex]\( z = y + 2 \)[/tex] (2 more pounds of walnuts than cashews)
### Step 3: Represent the equations as a matrix
We form an augmented matrix for these equations:
[tex]\[
\left[
\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}
\right]
\][/tex]
Here, [tex]\( x \)[/tex] represents pounds of almonds, [tex]\( y \)[/tex] represents pounds of cashews, and [tex]\( z \)[/tex] represents pounds of walnuts.
### Step 4: Solve the matrix
To solve for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex], we convert this matrix to reduced row echelon form (RREF). The RREF form will give us the solution directly.
### Step 5: Identify the solution
After converting the matrix to RREF and solving we get:
- [tex]\( x = 4 \)[/tex]
- [tex]\( y = 3 \)[/tex]
- [tex]\( z = 5 \)[/tex]
This means:
- The customer buys 4 pounds of almonds.
- The customer buys 3 pounds of cashews.
- The customer buys 5 pounds of walnuts.
### Step 6: Interpret the results
Now, let's interpret these results to match one of the provided statements:
- The customer buys 5 pounds of walnuts, 4 pounds of almonds, and 3 pounds of cashews.
- Compare walnuts to almonds: 5 - 4 = 1 (walnuts are 1 pound more than almonds)
- Compare almonds to cashews: 4 - 3 = 1 (almonds are 1 pound more than cashews)
So, the correct interpretation is:
- The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
Thus, the correct statement is:
The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
