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Amanda, Yolanda, and Rita sold items for a fundraiser. They sold cookies, candy, and cupcakes. The table below gives the number of boxes of each item sold by each person. They didn't sell any other items. If Amanda sold $57 in total, Yolanda sold $69, and Rita sold $68, what is the cost for a box of each item?

Let's denote the cost of one box of cookies as $ x $, the cost of one box of candy as $ y $, and the cost of one box of cupcakes as $ z $.

Using the information given in the table and the total amounts sold by each person, we can set up the following system of linear equations:

1. For Amanda:
\[
4x + 5y + 8z = 57
\]
2. For Yolanda:
\[
7x + 9y + 8z = 69
\]
3. For Rita:
\[
5x + 8y + 4z = 68
\]

Now, let's solve this system of equations to find the values of $ x $, $ y $, and $ z $.The solution to the system of equations gives us the following values:

- Cost of one box of cookies ($x$) = $-\frac{184}{9}$ dollars
- Cost of one box of candy ($y$) = $\frac{55}{3}$ dollars
- Cost of one box of cupcakes ($z$) = $\frac{53}{9}$ dollars

However, a negative value for the cost of cookies ($x$) doesn't make practical sense. This suggests there might be an inconsistency or error in the problem's setup or the given data. It could be worth re-examining the problem or checking for any possible mistakes.

Would you like to investigate this further, or do you have any specific questions about the solution?

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Here are some related questions you might consider:
1. How can we verify the correctness of the obtained solution?
2. What might cause a system of equations like this to give a non-physical (negative) result?
3. What techniques can be used to solve such systems of equations?
4. How could this problem be reformulated if a mistake in the table was suspected?
5. What other information might we need to correct this system?
6. How does changing one equation affect the solution?
7. Can we interpret the obtained solution in a different context?
8. How might the solution change if one of the totals was slightly different?

**Tip:** Always double-check the consistency of the data provided in mathematical problems, especially when dealing with real-world applications.

Learn how to solve a system of linear equations involving Amanda, Yolanda, and Rita's sales for a fundraiser. Understand the cost per box of cookies, candy, and cupcakes based on their total sales. Suitable for students in Grades 7-9.

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