Write a system of equations for this story problem please: A local grocery store makes a 9 pound mixture of trail mix. The trail mix contains raisins, sunflower seeds, and chocolate covered peanuts. The raisins cost $2 per pound, the sunflower seeds cost $1 per pound, and the chocolate covered peanuts cost $1.50 per pound. The mixture calls for twice as many raisins as sunflower seeds. The total cost of the mixture is $14.50.

Let's define the variables first:

- Let $ r $ represent the pounds of raisins.
- Let $ s $ represent the pounds of sunflower seeds.
- Let $ c $ represent the pounds of chocolate-covered peanuts.

Now, based on the information given in the problem, we can write the system of equations:

1. **Total weight of the mixture**: The total mixture weighs 9 pounds.
   \[
   r + s + c = 9
   \]
   
2. **Cost of the mixture**: The total cost is $14.50, where raisins cost $2 per pound, sunflower seeds cost $1 per pound, and chocolate-covered peanuts cost $1.50 per pound.
   \[
   2r + 1s + 1.5c = 14.50
   \]

3. **Twice as many raisins as sunflower seeds**: The problem states that the mixture contains twice as many raisins as sunflower seeds.
   \[
   r = 2s
   \]

So the system of equations is:

\[
\begin{aligned}
1. & \quad r + s + c = 9 \\
2. & \quad 2r + s + 1.5c = 14.50 \\
3. & \quad r = 2s
\end{aligned}
\]

Would you like to see the steps to solve this system?

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Here are 5 related questions to extend this problem:

1. How would the system change if the total cost increased to $18.50?
2. What would happen if the price of the sunflower seeds increased to $2 per pound?
3. What if the mixture required three times as many raisins as sunflower seeds?
4. How would you calculate the individual amounts of each ingredient if the total weight was only 7 pounds?
5. How could you check if this system has a unique solution?

**Tip**: When solving systems of equations, substitution is often useful when one equation expresses a variable in terms of another.

Learn how to solve a system of equations to determine the amounts of raisins, sunflower seeds, and chocolate-covered peanuts in a 9-pound trail mix. The problem includes concepts from algebra and linear equations, with a focus on substitution and cost analysis. Suitable for students in Grades 8-10.

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