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The problem asks for a system of equations and an augmented matrix that represent a situation involving pennies and nickels.

### Problem Breakdown:
- Charles has a total of 11 coins, consisting of **pennies (x)** and **nickels (y)**.
- The total value of the coins is $0.23, which is **23 cents**.
- Each penny is worth **0.01** dollars, and each nickel is worth **0.05** dollars.

### Step 1: Setting up the system of equations

We are given:
1. The total number of coins:  
   $ x + y = 11 $
   
2. The total value of the coins in dollars:  
   $ 0.01x + 0.05y = 0.23 $

So, the system of equations is:
\[
\begin{aligned}
x + y &= 11 \\
0.01x + 0.05y &= 0.23
\end{aligned}
\]

### Step 2: Creating the augmented matrix

To create the augmented matrix, write the coefficients of $ x $ and $ y $ from the system of equations, and the constant values on the right-hand side as follows:

\[
\begin{bmatrix}
1 & 1 & | & 11 \\
0.01 & 0.05 & | & 0.23
\end{bmatrix}
\]

This is the augmented matrix representing the system.

Would you like any more details on solving this system or further explanation? 

### 5 Related Questions:
1. How do you solve a system of equations using substitution or elimination?
2. How can augmented matrices be solved using Gaussian elimination?
3. What is the importance of using decimals instead of fractions in matrix representation?
4. How do you convert a system of equations into a matrix for larger systems (3x3, etc.)?
5. What is the difference between dependent, independent, and inconsistent systems of equations?

**Tip:** When solving a system of equations, double-check the units (cents vs. dollars) to ensure accuracy.

Charles has a collection of pennies and nickels worth $0.23. He has 11 coins. What are a system of equations and an augmented matrix that can represent this situation?

Solve the system of equations representing a collection of pennies and nickels, worth $0.23 and totaling 11 coins. This step-by-step solution includes the system of equations and the augmented matrix representation. Perfect for students in Grades 8-10.

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