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To solve this problem, we'll set up a system of equations based on the sales data given in each game. Let's denote:

- $ x $ as the price of a hot dog,
- $ y $ as the price of a hamburger,
- $ z $ as the price of a chicken sandwich.

### Game 1:
The concession stand sold 16 hot dogs, 14 hamburgers, and 8 chicken sandwiches for a total of $85.00. This can be expressed as:
\[
16x + 14y + 8z = 85
\]

### Game 2:
The concession stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $63.75. This can be expressed as:
\[
10x + 13y + 5z = 63.75
\]

### Game 3:
The concession stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $48.75. This can be expressed as:
\[
4x + 7y + 7z = 48.75
\]

We now have the following system of linear equations:
1. $ 16x + 14y + 8z = 85 $
2. $ 10x + 13y + 5z = 63.75 $
3. $ 4x + 7y + 7z = 48.75 $

### Solving the Equations
We'll solve these equations simultaneously to find the values of $ x $, $ y $, and $ z $.

Let's proceed:

1. **Equation 1**: $ 16x + 14y + 8z = 85 $
2. **Equation 2**: $ 10x + 13y + 5z = 63.75 $
3. **Equation 3**: $ 4x + 7y + 7z = 48.75 $

To simplify, let's use elimination or substitution methods to solve for the variables.

The solution to the system of equations is:

- The price of a hot dog ($ x $) is **$1.25**.
- The price of a hamburger ($ y $) is **$2.50**.
- The price of a chicken sandwich ($ z $) is **$3.75**.

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if the prices of one of the items increased by 50%?
2. Can you verify if these prices hold true for other combinations of sales quantities?
3. How can we use matrix methods (e.g., Gaussian elimination) to solve this system of equations?
4. What is the importance of having exactly three equations for three unknowns in solving such problems?
5. How could these equations change if there was an additional type of item being sold?

### Tip:
When dealing with systems of equations, always check if you have as many independent equations as unknowns to ensure a unique solution.

The concession stand at a school basketball tournament sells hot dogs, hamburgers, and chicken sandwiches. During one game, the stand sold 16 hot dogs, 14 hamburgers, and 8 chicken sandwiches for a total of $85.00. During a second game, the stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $63.75. During a third game, the stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $48.75. Determine the price of each product.

Learn how to determine the price of hot dogs, hamburgers, and chicken sandwiches using a system of linear equations. This problem involves three equations based on sales from different games and applies key algebraic concepts. Suitable for high school students in Grades 9-12.

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