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The problem reads:

> There are 15 plates in a kitchen cabinet. The diameter of each plate is either 7 inches or 12 inches. The diameter of all 15 plates combined is 140 inches. Which system of equations can be used to find the number of 7-inch plates, \( x \), and the number of 12-inch plates, \( y \), that are in the cabinet?

We are asked to:
1. Set up a system of equations based on this information.
2. Determine how many 7-inch plates are in the cabinet.

### Step 1: Setting Up Equations

Let:
- \( x \) = number of 7-inch plates
- \( y \) = number of 12-inch plates

Given:
- The total number of plates is 15.
  \[
  x + y = 15
  \]

- The total diameter is 140 inches.
  \[
  7x + 12y = 140
  \]

Thus, the system of equations is:
\[
\begin{cases} 
x + y = 15 \\
7x + 12y = 140 
\end{cases}
\]

### Step 2: Solving the System

1. From the first equation, solve for \( y \):
   \[
   y = 15 - x
   \]

2. Substitute \( y = 15 - x \) into the second equation:
   \[
   7x + 12(15 - x) = 140
   \]

3. Expand and simplify:
   \[
   7x + 180 - 12x = 140
   \]
   \[
   -5x = -40
   \]
   \[
   x = 8
   \]

4. Substitute \( x = 8 \) back into \( y = 15 - x \):
   \[
   y = 15 - 8 = 7
   \]

### Solution

- There are **8 plates** with a 7-inch diameter.
- There are **7 plates** with a 12-inch diameter.

Would you like more details on any step or have questions?

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### Related Questions

1. How would the solution change if there were a different total number of plates?
2. What if the total diameter were changed to 150 inches? How would we solve this?
3. How can you use substitution to solve similar word problems with systems of equations?
4. How would you verify if a solution to a system of equations is correct?
5. Can you represent this problem using a different method, such as graphing?

### Tip
In word problems involving systems of equations, identify the variables and equations based on constraints given in the problem.

There are 15 plates in a kitchen cabinet. The diameter of each plate is either 7 inches or 12 inches. The diameter of all 15 plates combined is 140 inches. Which system of equations can be used to find the number of 7-inch plates, x, and the number of 12-inch plates, y, that are in the cabinet? How many 7-inch plates are in the cabinet?

Learn how to solve a real-life word problem involving a system of equations. This problem finds the number of 7-inch and 12-inch plates in a cabinet given a total count and combined diameter. Suitable for students in Grades 8-10.

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