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The problem asks for a system of equations to find the price of a regular movie ticket, $ x $, and the price of a 3-D ticket, $ y $. The information provided in the question includes:

1. The price of a 3-D ticket is $2.00 more than a regular ticket, so:
   \[
   y = x + 2
   \]
   (This tells us that the 3-D ticket price is $2 more than the regular ticket price.)

2. Oliver bought five regular tickets and four 3-D tickets, and the total cost was $89. So, the total cost equation can be written as:
   \[
   5x + 4y = 89
   \]

Now, let's match this to the given options:

- **Option A**: $ x = y + 2 $ and $ 5x + 4y = 89 $ is incorrect because it reverses the relationship between $ x $ and $ y $.
- **Option B**: $ y = x + 2 $ and $ 4x + 5y = 89 $ is incorrect because the total cost equation should be $ 5x + 4y = 89 $, not $ 4x + 5y = 89 $.
- **Option C**: $ x + y = 2 $ and $ 4x + 5y = 89 $ is incorrect because the relationship between $ x $ and $ y $ should not be $ x + y = 2 $, and the total cost equation is also wrong.
- **Option D**: $ y = x + 2 $ and $ 5x + 4y = 89 $ is correct because it properly represents the relationship between the prices and the total cost equation.

Thus, the correct answer is **Option D**.

Would you like more details or have any further questions?

Here are five related questions to expand on this:

1. How would you solve this system of equations to find the exact values of $ x $ and $ y $?
2. What are the steps to substitute one equation into another when solving a system of equations?
3. How can we graphically interpret the solution to this system of equations?
4. What if the price difference between the 3-D and regular tickets was different? How would that affect the system of equations?
5. How can systems of equations be used in real-world financial decision-making?

**Tip**: When solving a system of equations, always make sure that each equation accurately reflects the relationships described in the problem!

Tickets to the 3-D showing of a certain movie cost $2.00 more than tickets to the regular showing. Oliver bought five regular tickets and four 3-D tickets, and the total cost was $89. Which system of equations can be used to determine the price of a regular ticket, x, and the price of a 3-D ticket, y?

This math problem explores how to solve a system of equations to determine the price of regular and 3-D movie tickets. Using algebra and substitution, find the correct system that describes the relationship between the two types of tickets, based on a total cost. Suitable for Grades 8-10.

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