https://file.aiquickdraw.com/mathbot/file/17266718879406f23gwj8.jpg

To model this problem, we will set up a system of three equations based on the given furniture combinations and their corresponding prices. We are tasked with determining the cost of a sofa (S), a love seat (L), and a chair (C).

### Step 1: Defining the Equations
From the image, the price combinations are as follows:

1. **Sofa and love seat cost $1300**:
   \[
   S + L = 1300
   \]

2. **Sofa and two chairs cost $1400**:
   \[
   S + 2C = 1400
   \]

3. **Sofa, love seat, and one chair cost $1600**:
   \[
   S + L + C = 1600
   \]

### Step 2: Solving the System of Equations
We now have the following system:

\[
S + L = 1300 \quad (1)
\]
\[
S + 2C = 1400 \quad (2)
\]
\[
S + L + C = 1600 \quad (3)
\]

Now, I will solve this system of equations to find the values of $ S $, $ L $, and $ C $. Let me calculate that for you.The solution to the system of equations is:

- The cost of a sofa $ S $ is **$800**.
- The cost of a love seat $ L $ is **$500**.
- The cost of a chair $ C $ is **$300**.

Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:
1. How would the system change if a sofa, love seat, and two chairs were priced together?
2. What would happen to the prices if the total price of the third combination increased by $200?
3. How would you solve this system graphically?
4. Can you determine the cost of 3 chairs and 2 love seats using the values of $ C $ and $ L $?
5. What would the solution look like if we knew only the total cost of one of the combinations?

**Tip:** When solving systems of equations, substitution or elimination methods are efficient ways to handle two or more variables.

Write a system of equations for the three combinations of furniture. Let S represent the cost of a sofa, L represent the cost of a love seat, and C represent the cost of a chair. The prices for the combinations are: $1300 for a sofa and a love seat, $1400 for a sofa and two chairs, and $1600 for a sofa, love seat, and one chair.

Learn how to model and solve a system of linear equations involving the pricing of furniture. This problem involves algebraic concepts such as linear equations and substitution or elimination methods. Suitable for students in Grades 8-10.

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