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Please solve in simple terms and show working 

Let's analyze and solve the problem shown in the image. 

### Problem Description:
- A commercial building contains some 1-bedroom and some 2-bedroom apartments.
- There are a total of 108 apartments.
- The total number of bedrooms in the building is 180.

### Goal:
- We need to find the number of 1-bedroom apartments and the number of 2-bedroom apartments.

### Steps to Solve:
Let's define:
- \( x \) = the number of 1-bedroom apartments
- \( y \) = the number of 2-bedroom apartments

We can set up the following system of equations based on the information provided:

1. **Total Apartments**:  
   Since there are 108 apartments in total, we have:
   \[
   x + y = 108
   \]

2. **Total Bedrooms**:  
   Each 1-bedroom apartment contributes 1 bedroom, and each 2-bedroom apartment contributes 2 bedrooms. Since the total number of bedrooms is 180, we can write:
   \[
   x + 2y = 180
   \]

### Solving the System of Equations:
We can solve this system of equations using substitution or elimination. Let's use the substitution method here:

1. From the first equation, solve for \( x \):
   \[
   x = 108 - y
   \]

2. Substitute \( x = 108 - y \) into the second equation:
   \[
   (108 - y) + 2y = 180
   \]

3. Simplify and solve for \( y \):
   \[
   108 + y = 180
   \]
   \[
   y = 180 - 108
   \]
   \[
   y = 72
   \]

4. Now, substitute \( y = 72 \) back into the equation \( x = 108 - y \) to find \( x \):
   \[
   x = 108 - 72
   \]
   \[
   x = 36
   \]

### Final Answer:
- The number of 1-bedroom apartments \( x \) is **36**.
- The number of 2-bedroom apartments \( y \) is **72**.

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. How would the equations change if the total number of apartments increased to 120?
2. What if each 1-bedroom apartment had 2 bedrooms instead? How would you set up the equations?
3. How would you solve this problem using the elimination method instead of substitution?
4. What if the total number of bedrooms was given as 200? How would that affect the solution?
5. Can you generalize the approach to solve for any number of total apartments and bedrooms?

### Tip:
For any word problem, defining variables clearly and setting up the correct equations based on the given conditions are key steps to finding the solution efficiently.

Learn how to solve a system of equations to determine the number of 1-bedroom and 2-bedroom apartments in a commercial building. This problem involves using algebraic methods to find solutions for variables representing different apartment types. Suitable for high school students and those interested in algebraic problem-solving.

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