You own a hamburger franchise and are planning to shut down operations for the day, but you are left with 13 buns, 19 defrosted beef patties, and 14 opened cheese slices. Rather than throw them out, you decide to use them to make burgers that you will sell at a discount. Plain burgers each require 1 beef patty and 1 bun, double cheeseburgers each require 2 beef patties, 1 bun, and 2 slices of cheese, while regular cheeseburgers each require 1 beef patty, 1 bun, and 1 slice of cheese. How many of each should you make?

plain burgers	  

double cheeseburgers	  

regular cheeseburgers	  



To solve this problem, we need to figure out how many plain burgers, double cheeseburgers, and regular cheeseburgers can be made using the available resources: 13 buns, 19 defrosted beef patties, and 14 opened cheese slices. We will set up a system of equations based on the requirements for each type of burger.

### Variables:
- Let \(x\) be the number of plain burgers.
- Let \(y\) be the number of double cheeseburgers.
- Let \(z\) be the number of regular cheeseburgers.

### Resource constraints:
1. **Buns constraint**:
   Each type of burger uses 1 bun. Thus, the total number of buns used is:
   \[
   x + y + z \leq 13
   \]
   
2. **Beef patties constraint**:
   - Plain burgers use 1 patty each.
   - Double cheeseburgers use 2 patties each.
   - Regular cheeseburgers use 1 patty each.
   The total number of patties used is:
   \[
   x + 2y + z \leq 19
   \]

3. **Cheese slices constraint**:
   - Plain burgers use no cheese.
   - Double cheeseburgers use 2 slices each.
   - Regular cheeseburgers use 1 slice each.
   The total number of cheese slices used is:
   \[
   2y + z \leq 14
   \]

### Objective:
We want to maximize the use of resources (buns, beef patties, and cheese slices) by determining how many of each type of burger should be made.

I'll now solve this system of inequalities.The solution to the system of equations is:

- \( x = 5 \) plain burgers,
- \( y = 6 \) double cheeseburgers,
- \( z = 2 \) regular cheeseburgers.

This combination maximizes the use of available buns, patties, and cheese slices.

Would you like a more detailed explanation, or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the solution change if we had more buns but the same amount of patties and cheese?
2. What if the cheese slices were reduced to 10, how would that affect the number of cheeseburgers?
3. Could we make more double cheeseburgers if we had 2 more patties?
4. How would the problem change if you wanted to leave a small inventory of buns and patties for tomorrow?
5. What other constraints could we add to further optimize this burger-making decision?

**Tip**: When solving problems with limited resources, setting up a system of equations or inequalities helps efficiently allocate resources.

You own a hamburger franchise and are planning to shut down operations for the day, but you are left with 13 buns, 19 defrosted beef patties, and 14 opened cheese slices. Rather than throw them out, you decide to use them to make burgers that you will sell at a discount. Plain burgers each require 1 beef patty and 1 bun, double cheeseburgers each require 2 beef patties, 1 bun, and 2 slices of cheese, while regular cheeseburgers each require 1 beef patty, 1 bun, and 1 slice of cheese. How many of each should you make?

Learn how to optimize burger production with 13 buns, 19 patties, and 14 cheese slices. This problem involves solving a system of linear inequalities to determine the number of plain burgers, double cheeseburgers, and regular cheeseburgers that can be made. Suitable for high school students in Grades 9-12.

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