The cost of attending an amusement park is $15 for children and $40 for adults. On a particular day, the attendance at the amusement park is 5,000 attendees, and the total money earned by the park is $100,000. 

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 this problem, we need to determine how many children and adult tickets were sold given the total number of attendees and total earnings.

### Step 1: Defining Variables
Let:
- $ x $ = the number of children's tickets sold
- $ y $ = the number of adult's tickets sold

### Step 2: Forming the System of Equations
We know:
- The total number of attendees is 5,000. Thus:
  \[
  x + y = 5000
  \]
- The total money earned is $100,000. Since each child’s ticket costs $15 and each adult’s ticket costs $40:
  \[
  15x + 40y = 100000
  \]

Now we have a system of two equations:
1. $ x + y = 5000 $
2. $ 15x + 40y = 100000 $

### Step 3: Matrix Representation

#### Coefficient Matrix
The coefficient matrix represents the coefficients of $ x $ and $ y $ in the system of equations. From the equations above, the coefficient matrix is:
\[
\begin{pmatrix}
1 & 1 \\
15 & 40
\end{pmatrix}
\]

#### Variable Matrix
The variable matrix contains the variables $ x $ and $ y $:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]

#### Solution Matrix
The solution matrix contains the constants on the right-hand side of the equations:
\[
\begin{pmatrix}
5000 \\
100000
\end{pmatrix}
\]

### Step 4: Complete Matrix Equation
The matrix equation combining these three matrices is:
\[
\begin{pmatrix}
1 & 1 \\
15 & 40
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
5000 \\
100000
\end{pmatrix}
\]

This is the matrix setup to solve for $ x $ (number of children's tickets) and $ y $ (number of adult's tickets).

Would you like me to solve this system of equations next?

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Here are 5 related questions:
1. How can this system of equations be solved using substitution or elimination methods?
2. What is the inverse of the coefficient matrix, and how can it be used to solve the system?
3. If the ticket prices change, how would the system of equations be affected?
4. How can matrices be used to solve systems of equations with more than two variables?
5. What other real-world problems can be modeled with a similar matrix system?

**Tip:** When solving systems of equations with matrices, ensure the coefficient matrix is invertible to guarantee a unique solution.

Learn how to solve a system of equations using matrices to find the number of children's and adult's tickets sold at an amusement park. This problem involves applying matrix algebra to determine ticket sales based on total attendees and revenue. Suitable for high school students studying systems of linear equations.

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