https://file.aiquickdraw.com/mathbot/file/1726586332699rzxdm5f7.jpg

Let's break down the problem and form the system of equations.

### Given:
- A popsicle is sold for $5.
- A hot dog is sold for $7.
- A hamburger is sold for $9.
- A total of 255 items were sold.
- Total revenue = $1931.
- They sold twice as many hamburgers as popsicles.

### Variables:
- Let $ x $ be the number of popsicles sold.
- Let $ y $ be the number of hot dogs sold.
- Let $ z $ be the number of hamburgers sold.

### Equations:

1. **Total sales equation**: 
   \[
   x + y + z = 255
   \]
   (Since the total number of items sold is 255.)

2. **Total revenue equation**: 
   \[
   5x + 7y + 9z = 1931
   \]
   (Each item sold has a price, and the total revenue is $1931.)

3. **Relation between hamburgers and popsicles**: 
   \[
   z = 2x
   \]
   (Twice as many hamburgers were sold as popsicles.)

### System of equations:
\[
\begin{aligned}
1. \quad & x + y + z = 255 \\
2. \quad & 5x + 7y + 9z = 1931 \\
3. \quad & z = 2x
\end{aligned}
\]

These are the three equations needed.

Would you like to solve this system, or do you need further clarification? Here are five related questions you could explore:

1. How would you solve this system of equations step by step?
2. How does the condition $ z = 2x $ simplify the system?
3. What happens if the prices of the items change?
4. Can you create a similar problem with different constraints or relationships?
5. How would graphing these equations look like in two or three dimensions?

**Tip**: Always substitute known relationships (like $ z = 2x $) into other equations to reduce the number of variables.

A food truck sells three types of food: a popsicle for $5, a hot dog for $7, and a hamburger for $9. Yesterday, the food truck made a total of 255 sales for a total of $1931. They sold twice as many hamburgers as popsicles. Write the equations to find the number of each type of hot dogs, popsicles, and hamburgers sold. Let x be the number of popsicles sold, y be the number of hot dogs sold, and z be the number of hamburgers sold. Do not solve.

Explore how to write a system of equations to represent a food truck's sales problem, involving items like popsicles, hot dogs, and hamburgers. The problem includes 255 total sales and revenue of $1931, using algebraic methods to form a set of linear equations.

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