Problem 55 - College Algebra

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule. A movie theater needs to know how many adult tickets and children tickets were sold out of the \(1,200\) total tickets. If children's tickets are \(\$ 5.95\) adult tickets are \(\$ 11.15\) , and the total amount of revenue was \(\$ 12,756\) , how many children's tickets and adult tickets were sold?

Step-by-Step Solution

Answer

Approximately 137 adult tickets and 936 children's tickets were sold.

Step 1: Identify Variables

Let \( x \) represent the number of adult tickets and \( y \) represent the number of children's tickets sold. We need to find values for \( x \) and \( y \).

Step 2: Formulate Equations

We have two pieces of information: 1) The total number of tickets, and 2) The total revenue from ticket sales. Thus, we get the equations: \[ x + y = 1200 \] and \[ 11.15x + 5.95y = 12756 \]

Step 3: Set Up the Coefficient Matrix

The coefficient matrix \( A \) is created from the coefficients of the \( x \) and \( y \) terms in the equations: \[ A = \begin{bmatrix} 1 & 1 \ 11.15 & 5.95 \end{bmatrix} \]

Step 4: Calculate Determinant of Coefficient Matrix

The determinant \( \text{det}(A) \) is calculated as follows: \[ \text{det}(A) = (1)(5.95) - (1)(11.15) = -5.2 \]

Step 5: Calculate Determinant for Cramer’s Rule (X-Matrix)

Replace the first column of \( A \) with the constant terms from the equations to create \( A_x \): \[ A_x = \begin{bmatrix} 1200 & 1 \ 12756 & 5.95 \end{bmatrix} \] Find the determinant of \( A_x \): \[ \text{det}(A_x) = (1200)(5.95) - (1)(12756) = 714 \]

Step 6: Calculate Determinant for Cramer’s Rule (Y-Matrix)

Replace the second column of \( A \) with the constant terms from the equations to create \( A_y \): \[ A_y = \begin{bmatrix} 1 & 1200 \ 11.15 & 12756 \end{bmatrix} \] Find the determinant of \( A_y \): \[ \text{det}(A_y) = (1)(12756) - (11.15)(1200) = 486 \]

Step 7: Apply Cramer’s Rule

Cramer's rule states that \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \) and \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \). Compute \( x \) and \( y \): \[ x = \frac{714}{-5.2} \approx 137 \] \[ y = \frac{486}{-5.2} \approx 936 \]

Step 8: Interpret Solution

The approximate solution indicates that about 137 adult tickets and 936 children's tickets were sold, maintaining the total number of tickets and revenue constraints.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In the exercise, the theater's ticket sales form a system of equations. Each equation represents a relationship, such as the total number of tickets sold and the total revenue generated. For instance, the equations from our problem are:

• \( x + y = 1200 \), where \( x \) is the number of adult tickets, and \( y \) is the number of children's tickets.
• \( 11.15x + 5.95y = 12756 \), indicating the total revenue.

These equations are interdependent and need to be solved simultaneously to find the exact number of each type of ticket sold.

Coefficient Matrix

The coefficient matrix is crucial in solving a system of equations using matrix methods, like Cramer's Rule. It is a matrix that consists of only the coefficients of the variables in the linear equations. In our exercise, the coefficient matrix derived from the system is:\[A = \begin{bmatrix} 1 & 1 \ 11.15 & 5.95 \end{bmatrix}\]This matrix helps in organizing the information neatly, making it easier to apply matrix operations to solve the equations. Notice the matrix does not include the constant terms from the right side of the equations. That ensures the focus remains on the relationship dictated by the coefficients.

Determinant

The determinant is a special number calculated from a square matrix, which can give insights regarding the matrix's properties, such as invertibility. In solving systems of linear equations via Cramer's Rule, the determinant helps in finding solutions to the variablesFor the coefficient matrix \( A \) in this problem, the determinant is calculated as:\[\text{det}(A) = (1)(5.95) - (1)(11.15) = -5.2\]The determinant must not be zero, as a zero determinant indicates the system of equations may not have a unique solution. In this exercise, the non-zero determinant confirms the system has a unique solution.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vector spaces and linear mappings between these spaces. This includes concepts such as vectors, matrices, and linear equations.
In the context of this exercise, linear algebra allows us to apply methods like Cramer's Rule to efficiently solve for variables in a system of linear equations.
Cramer's Rule is just one among many techniques available to solve linear systems. By utilizing determinants and matrices, linear algebra provides a robust framework for understanding and solving complex systems of equations, making tasks like optimizing ticket sales analysis both flexible and computationally feasible.