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 the problem of determining the number of children's and adults' tickets sold, we can set up a system of linear equations based on the information given. The goal is to find the number of children's tickets $x$ and the number of adults' tickets $y$ sold. 

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

### Step 2: Set Up the System of Equations
We have two key pieces of information:
1. The total number of attendees: $ x + y = 5000 $
2. The total money earned: $ 15x + 40y = 100,000 $

### Step 3: Write the System of Equations
\[
\begin{aligned}
1. & \quad x + y = 5000 \quad &\text{(Equation 1)} \\
2. & \quad 15x + 40y = 100,000 \quad &\text{(Equation 2)}
\end{aligned}
\]

### Step 4: Set Up the Matrices
We can express this system of equations in matrix form as follows:

#### Coefficient Matrix (A):
\[
A = \begin{pmatrix}
1 & 1 \\
15 & 40
\end{pmatrix}
\]

This matrix contains the coefficients of $ x $ and $ y $ from the system of equations.

#### Variable Matrix (X):
\[
X = \begin{pmatrix}
x \\
y
\end{pmatrix}
\]

This matrix contains the variables $ x $ and $ y $ that we are trying to solve for.

#### Solution Matrix (B):
\[
B = \begin{pmatrix}
5000 \\
100,000
\end{pmatrix}
\]

This matrix contains the constants from the right-hand side of the system of equations.

### Step 5: Matrix Equation
The system of equations can now be written as a matrix equation:
\[
AX = B
\]
Which translates to:
\[
\begin{pmatrix}
1 & 1 \\
15 & 40
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
5000 \\
100,000
\end{pmatrix}
\]

### Explanation
- The coefficient matrix $ A $ represents the coefficients of the variables $ x $ and $ y $ in the equations.
- The variable matrix $ X $ contains the unknowns, which are the number of children's and adults' tickets sold.
- The solution matrix $ B $ represents the total number of attendees and the total revenue.

This matrix equation can be solved using methods such as substitution, elimination, or matrix operations (like finding the inverse of $ A $ and multiplying both sides by it) to determine the values of $ x $ and $ y $.

Would you like the detailed steps for solving this matrix equation, or do you have any other questions?

### Related Questions
1. How do you solve the system of equations using substitution?
2. How do you solve the system of equations using the elimination method?
3. What is the determinant of the coefficient matrix, and how does it affect the solution?
4. How do you find the inverse of a matrix, and how is it used to solve a system of equations?
5. Can you solve the system of equations using Cramer's Rule?
6. What happens if the determinant of the coefficient matrix is zero?
7. How would the system change if the total revenue was different?
8. What if the cost of children's tickets increased by $5? How would that affect the solution?

### Tip
When setting up a system of equations, always double-check the equations to ensure they accurately represent the problem's conditions before proceeding to solve them.

Learn how to calculate the number of children's and adults' tickets sold at an amusement park using linear equations. Understand the matrix setup and solve for attendance and revenue. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation