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We are given three sets of purchases:

1. **12 balloons**, **7 cupcakes**, and **8 hats** cost **$49**.
2. **8 balloons**, **8 cupcakes**, and **7 hats** cost **$46**.
3. **11 balloons**, **9 cupcakes**, and **9 hats** cost **$56**.

We need to write a system of equations representing this situation where:
- $ b $ is the number of balloons,
- $ c $ is the number of cupcakes,
- $ h $ is the number of hats.

The equations are derived based on the cost structure of each set:

1. $ 12b + 7c + 8h = 49 $
2. $ 8b + 8c + 7h = 46 $
3. $ 11b + 9c + 9h = 56 $

Now, let's match these equations to the given options:

### Checking Option B:
- First equation: $ 12b + 7c + 8h = 49 $ ✔️ Matches.
- Second equation: $ 8b + 8c + 7h = 46 $ ✔️ Matches.
- Third equation: $ 11b + 9c + 9h = 56 $ ✔️ Matches.

So, **Option B** is the correct choice.

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Do you want further details or have any questions?

Here are 5 related questions for practice:
1. How can you solve this system of equations using substitution?
2. What is the value of $ b $ (balloons) if the price of a cupcake and a hat is known?
3. How can you represent the total cost equation for a different combination of cupcakes, balloons, and hats?
4. How would the system of equations change if the price of cupcakes were reduced by $1 for each set?
5. What is the determinant of the coefficient matrix in this system?

**Tip:** When solving a system of equations, it's often helpful to simplify the coefficients before using elimination or substitution to reduce potential errors.

Tinisha is planning a birthday party. She can buy: 12 balloons, 7 cupcakes, and 8 hats for $49; 8 balloons, 8 cupcakes, and 7 hats for $46; or 11 balloons, 9 cupcakes, and 9 hats for $56. Using b for the number of balloons, c for the number of cupcakes, and h for the number of hats, write a system of equations to represent this situation.

Learn how to represent a real-world problem using a system of linear equations. In this example, Tinisha buys balloons, cupcakes, and hats for a party, and the goal is to form a system of equations based on her purchase. This is ideal for students in Grades 8-10.

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