A store manager mixes two kinds of nuts, one worth $1.45/pound and the other worth $1.75/pound.  She wants to make a 36 pound mixture that will be sold for $1.55/pound.

a. Set up a system of two equations with two unknowns to summarize this information.

To solve this problem, we need to set up a system of two equations based on the given information. Let's define the variables as follows:

- Let $ x $ be the number of pounds of the first type of nut (worth $1.45 per pound).
- Let $ y $ be the number of pounds of the second type of nut (worth $1.75 per pound).

### The two equations can be derived from the following considerations:

1. **Total Weight Equation:**  
   The total weight of the mixture is 36 pounds. So, we have:
   \[
   x + y = 36
   \]

2. **Total Value Equation:**  
   The total value of the mixture must equal the price per pound of the mixture times the total weight, which is $ 1.55 \times 36 $ dollars. This can be expressed as:
   \[
   1.45x + 1.75y = 1.55 \times 36
   \]

So, the system of equations summarizing this information is:

\[
\begin{aligned}
x + y &= 36 \quad \text{(1)}\\
1.45x + 1.75y &= 1.55 \times 36 \quad \text{(2)}
\end{aligned}
\]

Would you like to see the next steps to solve this system, or have any questions? 

Here are 8 related questions to deepen your understanding:

1. What is the solution to this system of equations?
2. How do you solve the system of equations using substitution?
3. How do you solve the system of equations using the elimination method?
4. What is the value of $ 1.55 \times 36 $?
5. How many pounds of each type of nut are in the mixture?
6. How would the system change if the desired price per pound was $1.60?
7. How does the cost per pound of the nuts affect the mixture?
8. How would the system change if the total mixture was 50 pounds instead of 36 pounds?

**Tip:** When setting up systems of equations, make sure that each equation accurately represents a different aspect of the problem—such as total quantity and total cost.

Learn how to solve a problem where a store manager mixes two types of nuts, priced at $1.45/pound and $1.75/pound, to create a 36 pound mixture sold at $1.55/pound. This problem involves setting up and solving a system of linear equations, suitable for Grades 7-9.

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