A store is having a sale on chocolate chips and walnuts. For 2 pounds of chocolate chips and 3 pounds of walnuts, the total cost is $11. For 6 pounds of chocolate chips and 5 pounds of walnuts, the total cost is $23. Find the cost for each pound of chocolate chips and each pound of walnuts.

To solve this problem, we can set up a system of linear equations based on the information provided.

Let's define:
- $ x $ as the cost per pound of chocolate chips.
- $ y $ as the cost per pound of walnuts.

### Step 1: Set up the equations
From the problem statement, we have two conditions:

1. For 2 pounds of chocolate chips and 3 pounds of walnuts, the total cost is $11:
   \[
   2x + 3y = 11
   \]
   
2. For 6 pounds of chocolate chips and 5 pounds of walnuts, the total cost is $23:
   \[
   6x + 5y = 23
   \]

### Step 2: Solve the system of equations

We can solve this system using either substitution or elimination. Let's use the elimination method.

1. Multiply the first equation by 2.5 to align the coefficients of $ y $:
   \[
   2.5(2x + 3y) = 2.5(11)
   \]
   This simplifies to:
   \[
   5x + 7.5y = 27.5
   \]

2. Multiply the second equation by 1.5:
   \[
   1.5(6x + 5y) = 1.5(23)
   \]
   This simplifies to:
   \[
   9x + 7.5y = 34.5
   \]

3. Subtract the first modified equation from the second modified equation:
   \[
   (9x + 7.5y) - (5x + 7.5y) = 34.5 - 27.5
   \]
   Simplifying this gives:
   \[
   4x = 7
   \]
   \[
   x = \frac{7}{4} = 1.75
   \]

### Step 3: Substitute the value of $ x $ back into one of the original equations

Substitute $ x = 1.75 $ into the first equation:
\[
2(1.75) + 3y = 11
\]
\[
3.5 + 3y = 11
\]
\[
3y = 7.5
\]
\[
y = \frac{7.5}{3} = 2.5
\]

### Final Answer:
- The cost per pound of chocolate chips is $ \boxed{1.75} $ dollars.
- The cost per pound of walnuts is $ \boxed{2.5} $ dollars.

Would you like more details or have any questions?

### Related Questions:
1. How can you verify that the solution is correct by substituting the values back into the original equations?
2. What would the total cost be for 4 pounds of chocolate chips and 2 pounds of walnuts?
3. How would the system of equations change if the total cost for the second scenario was different, say $25 instead of $23?
4. What method can be used to solve this problem if the system of equations was more complex, involving three variables?
5. Can you explain how to solve the system using the substitution method instead of elimination?
6. What is the graphical interpretation of the solution to this system of equations?
7. If the price of walnuts increased by 10%, how would this affect the solution?
8. How can this system of equations be solved using matrix operations?

### Tip:
When dealing with word problems, clearly defining your variables and setting up the correct equations is key to finding the solution.

Learn how to solve a system of linear equations involving chocolate chips and walnuts. Understand the method of elimination to find the cost per pound of each item based on given total costs and quantities. Suitable for students in Grades 7-9.

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