The cost of attending an amusement park is $10 for children and $20 for adults. On a particular day, the attendance at the amusement park is 30,000 attendees, and the total money earned by the park is $500,000.

Set up the coefficient matrix, variable matrix, and solution matrix that could be used to determine the number of children’s and adult’s tickets sold. Show or explain all necessary steps.




To solve the problem of determining the number of children's and adult's tickets sold, we can set up a system of linear equations based on the information given:

### Step 1: Define Variables
- Let $ x $ represent the number of children's tickets sold.
- Let $ y $ represent the number of adult's tickets sold.

### Step 2: Set Up the Equations
Given the information:
1. The total number of attendees is 30,000.
   \[
   x + y = 30,000
   \]
2. The total money earned is $500,000.
   \[
   10x + 20y = 500,000
   \]

### Step 3: Set Up the Coefficient Matrix, Variable Matrix, and Solution Matrix
These equations can be represented in matrix form as:

1. **Coefficient Matrix**:
   \[
   \mathbf{A} = \begin{pmatrix} 
   1 & 1 \\ 
   10 & 20 
   \end{pmatrix}
   \]

2. **Variable Matrix**:
   \[
   \mathbf{X} = \begin{pmatrix} 
   x \\ 
   y 
   \end{pmatrix}
   \]

3. **Solution Matrix**:
   \[
   \mathbf{B} = \begin{pmatrix} 
   30,000 \\ 
   500,000 
   \end{pmatrix}
   \]

### Step 4: Matrix Equation
The system of equations can be written as a single matrix equation:

\[
\mathbf{A} \mathbf{X} = \mathbf{B}
\]

Which is:

\[
\begin{pmatrix} 
1 & 1 \\ 
10 & 20 
\end{pmatrix}
\begin{pmatrix} 
x \\ 
y 
\end{pmatrix}
=
\begin{pmatrix} 
30,000 \\ 
500,000 
\end{pmatrix}
\]

This matrix equation represents the system of linear equations that can be solved to find the values of $ x $ and $ y $, which are the number of children's and adult's tickets sold, respectively.

Would you like to see the solution to this system, or do you have any other questions?

### Related Questions

1. How can you solve the matrix equation $\mathbf{A} \mathbf{X} = \mathbf{B}$ using the inverse of the matrix $\mathbf{A}$?
2. What are the conditions for the existence of a unique solution in a system of linear equations?
3. How would the equations change if the price for adult tickets was different?
4. Can the system of equations be solved using the substitution method? How?
5. What would the system of equations look like if the park had 40,000 attendees instead?
6. How can you interpret the determinant of the coefficient matrix $\mathbf{A}$?
7. What if the park made $600,000 in a day with 30,000 attendees? How would that change the matrix equation?
8. Can you solve this system graphically? What does the intersection point represent?

#### Tip:
When working with matrices, ensure that the matrix $\mathbf{A}$ is invertible (i.e., it has a non-zero determinant) before attempting to solve the system using the inverse matrix method.

Explore how to solve the problem of determining the number of children's and adult's tickets sold at an amusement park using matrix equations. Understand the application of linear equations and matrix operations in real-world scenarios. Suitable for high school students and anyone interested in applied mathematics.

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