You want to order a deli platter for a sports banquet. You need 12 pounds of meat (ham, turkey, and roast beef) and cheese total. You want twice as much meat as cheese on the platter. You wants the same amount of ham and turkey. The price per pound is $4.95 for ham, $6.99 for turkey, $7.99 for roast beef, and $4.36 for cheese. How many punds of each should you order if you plan to spend $69.24?

We are tasked with determining how many pounds of each ingredient (ham, turkey, roast beef, and cheese) you should order, given specific conditions. Let's break it down step by step.

### Define the variables:
Let:
- $ h $ be the number of pounds of ham.
- $ t $ be the number of pounds of turkey.
- $ r $ be the number of pounds of roast beef.
- $ c $ be the number of pounds of cheese.

From the problem, we know:

1. **Total weight of meat and cheese**:  
   You need 12 pounds of meat and cheese combined.  
   Since the total amount of meat and cheese must equal 12 pounds, we can write:
   \[
   h + t + r + c = 12
   \]

2. **Meat is twice as much as cheese**:  
   You want twice as much meat (ham, turkey, roast beef) as cheese.  
   Thus, the amount of meat $ h + t + r $ should be twice the amount of cheese $ c $, so:
   \[
   h + t + r = 2c
   \]

3. **Equal amounts of ham and turkey**:  
   The amount of ham is equal to the amount of turkey, so:
   \[
   h = t
   \]

4. **Total cost**:  
   The total cost is $69.24, where the price per pound is $4.95 for ham, $6.99 for turkey, $7.99 for roast beef, and $4.36 for cheese. The total cost can be written as:
   \[
   4.95h + 6.99t + 7.99r + 4.36c = 69.24
   \]

### Step-by-step solution:

#### Step 1: Use the information that $ h = t $
Substitute $ t = h $ into the equations.

#### Step 2: Simplify the system of equations:
From $ h + t + r + c = 12 $, substitute $ t = h $:
\[
h + h + r + c = 12 \quad \Rightarrow \quad 2h + r + c = 12
\]

From $ h + t + r = 2c $, substitute $ t = h $:
\[
h + h + r = 2c \quad \Rightarrow \quad 2h + r = 2c \quad \Rightarrow \quad r = 2c - 2h
\]

#### Step 3: Substitute $ r $ in terms of $ h $ and $ c $ into the cost equation:
The cost equation is:
\[
4.95h + 6.99t + 7.99r + 4.36c = 69.24
\]
Substitute $ t = h $ and $ r = 2c - 2h $:
\[
4.95h + 6.99h + 7.99(2c - 2h) + 4.36c = 69.24
\]
Simplify the equation:
\[
(4.95 + 6.99)h + 7.99(2c - 2h) + 4.36c = 69.24
\]
\[
11.94h + 15.98c - 15.98h + 4.36c = 69.24
\]
\[
-4.04h + 20.34c = 69.24
\]

#### Step 4: Solve the system of equations:
We now have two equations:
1. $ 2h + r + c = 12 $
2. $ -4.04h + 20.34c = 69.24 $

Solve this system to find $ h $, $ r $, and $ c $.The solution to the system of equations is as follows:

- $ h = 3 $ pounds of ham,
- $ t = 3 $ pounds of turkey (since $ h = t $),
- $ r = 2 $ pounds of roast beef,
- $ c = 4 $ pounds of cheese.

Thus, you should order:
- 3 pounds of ham,
- 3 pounds of turkey,
- 2 pounds of roast beef,
- 4 pounds of cheese.

This satisfies both the total weight and the cost requirements.

Would you like further details or have any questions?

Here are some related questions to explore:
1. How would the order change if the price per pound of roast beef increased?
2. What if the ratio of meat to cheese was different, like 3:1?
3. How would the solution change if you wanted more roast beef than ham and turkey?
4. What would be the total cost if you only ordered ham and cheese?
5. How would your answer change if you had a higher budget?

**Tip:** When solving systems of equations with multiple variables, expressing one variable in terms of others can simplify the process significantly.

You want to order a deli platter for a sports banquet. You need 12 pounds of meat (ham, turkey, and roast beef) and cheese total. You want twice as much meat as cheese on the platter. You want the same amount of ham and turkey. The price per pound is $4.95 for ham, $6.99 for turkey, $7.99 for roast beef, and $4.36 for cheese. How many pounds of each should you order if you plan to spend $69.24?

Solve a real-world algebra problem involving ordering a deli platter with 12 pounds of meat and cheese, while maintaining a budget of $69.24. Learn to calculate the amounts of ham, turkey, roast beef, and cheese using algebraic equations. Suitable for students in Grades 9-11.

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