Problem 63 - 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 theatre sold tickets to three movies. The tickets to the first movie were \(5, the tickets to the second movie were \)11, and the third movie was \(12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of \)6,774. How many tickets for each movie were sold?

Step-by-Step Solution

Answer

100 tickets for the first movie, 230 for the second, and 312 for the third.

Step 1: Define Variables

Let \( x \) represent the number of tickets sold for the first movie, \( y \) represent the number of tickets sold for the second movie, and \( z \) represent the number of tickets sold for the third movie.

Step 2: Set Up Equations from the Information Given

We have three pieces of information leading to three equations:1. 100 tickets were sold to the first movie: \( x = 100 \).2. Total tickets sold are 642: \( x + y + z = 642 \).3. Total revenue is \$6,774: \( 5x + 11y + 12z = 6774 \).

Step 3: Substitute Known Quantity

Substitute \( x = 100 \) into the second and third equations:1. \( 100 + y + z = 642 \) simplifies to \( y + z = 542 \).2. \( 5 \times 100 + 11y + 12z = 6774 \) simplifies to \( 11y + 12z = 6274 \).

Step 4: Set Up Matrix for Cramer's Rule

Write the system of equations in matrix form:1. \( \begin{bmatrix} 1 & 1 \ & 11 & 12 \end{bmatrix} \begin{bmatrix} y \ z \end{bmatrix} = \begin{bmatrix} 542 \ 6274 \end{bmatrix} \)

Step 5: Calculate Determinants for Cramer's Rule

1. Calculate the determinant of the coefficient matrix: \( \text{det}(A) = \begin{vmatrix} 1 & 1 \ 11 & 12 \end{vmatrix} = (1)(12) - (1)(11) = 1 \).2. Calculate the determinants for each of the matrices needed for Cramer's Rule:- \( \text{det}(A_y) = \begin{vmatrix} 542 & 1 \ 6274 & 12 \end{vmatrix} = (542)(12) - (1)(6274) = 6504 - 6274 = 230 \).- \( \text{det}(A_z) = \begin{vmatrix} 1 & 542 \ 11 & 6274 \end{vmatrix} = (1)(6274) - (542)(11) = 6274 - 5962 = 312 \).

Step 6: Solve for Each Variable Using Cramer’s Rule

Using Cramer's Rule:\( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{230}{1} = 230 \) \( z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{312}{1} = 312 \)

Step 7: Review the Solution

The number of tickets sold for each movie is:- First movie: \( x = 100 \)- Second movie: \( y = 230 \)- Third movie: \( z = 312 \)

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations that share the same set of variables. In the context of this ticket sales problem, we are dealing with three linear equations involving the variables \( x, y, \text{and } z \), which represent the number of tickets sold for each of the three movies respectively.

The goal here is to find values of these variables that satisfy all the equations simultaneously. The given information allows us to set up three equations:

• \( x = 100 \) because 100 tickets were sold for the first movie.
• \( x + y + z = 642 \) represents the total number of tickets sold for all the movies combined.
• \( 5x + 11y + 12z = 6774 \) accounts for the total revenue from the ticket sales, considering each movie ticket's different price.

Once these equations are established, they can be solved together to determine the exact number of tickets sold for each individual movie.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements and provides important information about the matrix. It is essential in solving systems of linear equations using Cramer's Rule.

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed as \( \, ad - bc \). In our ticket sales problem, once we substituted \( x = 100 \), the two remaining equations from the system were transformed into matrix form to allow the use of Cramer’s Rule.

• The coefficient matrix \( \begin{bmatrix} 1 & 1 \ 11 & 12 \end{bmatrix} \) had a determinant \( \text{det}(A) = (1)(12) - (1)(11) = 1 \).
• This simple determinant calculation is what enables the application of Cramer's Rule efficiently, as a non-zero determinant indicates that the system has a unique solution.

Calculating other determinants, such as \( \text{det}(A_y) \) and \( \text{det}(A_z) \), involves replacing the appropriate columns with constants from the equations to find specific variable solutions, confirming the unique ticket sale distribution among the movies.

Ticket Sales Problem

The ticket sales problem is a practical application of systems of linear equations. It demonstrates how such equations can be used to solve real-world problems involving finance and accounting.

This particular problem involved different pricing for tickets across three movies, with sales accumulating to a known total revenue. By establishing equations that incorporated these prices and totals, we could use mathematical methods like Cramer's Rule to accurately derive ticket numbers sold for each movie.

Once put into the context of matrices and determinants, what might seem like an abstract calculation becomes a logical solution path to a concrete question: determining how audiences distributed their viewership among the three films. By solving such problems, students develop critical-thinking skills and gain insights into mathematical models' effectiveness in operational analysis and planning.