Problem 55 - College Algebra with Corequisite Support

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

1080 adult and 120 children's tickets were sold.

Step 1: Understand the Problem

We need to find the number of adult tickets and children's tickets sold. We know the total number of tickets sold is 1,200, the price of each type of ticket, and the total revenue collected.

Step 2: Define Variables

Let \(x\) represent the number of adult tickets sold and \(y\) represent the number of children's tickets sold.

Step 3: Create the System of Equations

We have two key pieces of information: 1. The total number of tickets sold: \[ x + y = 1200 \]2. The total revenue: \[ 11.15x + 5.95y = 12756 \]

Step 4: Organize for Cramer's Rule

To use Cramer's Rule, we need to arrange our equations in matrix form, which is:\[ \begin{bmatrix} 1 & 1 \ 11.15 & 5.95 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1200 \ 12756 \end{bmatrix} \]

Step 5: Calculate the Determinant of the Coefficient Matrix

The determinant \(D\) of the coefficient matrix is calculated as:\[ D = \begin{vmatrix} 1 & 1 \ 11.15 & 5.95 \end{vmatrix} = (1)(5.95) - (1)(11.15) = 5.95 - 11.15 = -5.2 \]

Step 6: Calculate Determinants for Cramer's Rule

Calculate the determinant for each variable:\[ D_x = \begin{vmatrix} 1200 & 1 \ 12756 & 5.95 \end{vmatrix} = (1200)(5.95) - (1)(12756) = 7140 - 12756 = -5616 \]\[ D_y = \begin{vmatrix} 1 & 1200 \ 11.15 & 12756 \end{vmatrix} = (1)(12756) - (11.15)(1200) = 12756 - 13380 = -624 \]

Step 7: Solve for the Variables Using Cramer's Rule

Using Cramer's Rule, solve for \(x\) and \(y\):\[ x = \frac{D_x}{D} = \frac{-5616}{-5.2} = 1080 \]\[ y = \frac{D_y}{D} = \frac{-624}{-5.2} = 120 \]

Step 8: Verify the Solution

Double-check our solutions by plugging them back into the original equations:1. \( x + y = 1080 + 120 = 1200 \) (Correct)2. \( 11.15x + 5.95y = 11.15(1080) + 5.95(120) = 12756 \) (Correct)

Key Concepts

Linear Equations

Linear equations are fundamental mathematical expressions where each term is either a constant or the product of a constant and a single variable. These equations are termed "linear" because, when graphed, they form a straight line. A typical linear equation in two variables looks like this: \( ax + by = c \), where:

• \( x \) and \( y \) are variables, representing unknowns to be solved.
• \( a \) and \( b \) are coefficients, which are known values multiplied by the variables.
• \( c \) is a constant, a known value.

In the context of the movie theater problem, our linear equations represent real-world constraints. The first linear equation, \( x + y = 1200 \), communicates that the total number of tickets sold was 1,200.
The second equation, \( 11.15x + 5.95y = 12756 \), indicates the total revenue. These linear equations help us model and solve complex real-world situations by simplifying them into understandable patterns.

System of Equations

A system of equations consists of two or more equations with the same set of variables. Solving a system means finding the values of the variables that satisfy every equation simultaneously. In practical scenarios, such as our movie ticket problem, systems of equations help us solve for multiple unknowns. We use this approach when we have more than one condition, such as total tickets and total revenue, that must both be met.In this exercise, the system of equations is:

• \( x + y = 1200 \)
• \( 11.15x + 5.95y = 12756 \)

Solving the system arrives at the solution: 1,080 adult tickets and 120 children's tickets. Each equation in the system is representing a real constraint, and by solving the system, we find consistent numbers that satisfy both equations.

Determinants

Determinants are special numbers calculated from square matrices, which have equal rows and columns. They provide vital insight into various properties of matrices, such as the matrix's invertibility. The determinant of a 2x2 matrix, for example, is calculated using the formula:\[D = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc\]In the Cramer’s Rule solution, determinant calculations allow us to find solutions to the system of equations. For the coefficient matrix and each variable's matrix, we compute determinants to discover if the system has a unique solution. Here:

• The determinant of the coefficient matrix \( D \) is \(-5.2\).
• The determinant for \( x \), \( D_x \), is \(-5616\).
• The determinant for \( y \), \( D_y \), is \(-624\).

The approach of using determinants in systems of equations elegantly encapsulates complex calculations into straightforward operations, providing a path to solutions.

Matrix Form

Matrix form is a structured way to represent systems of linear equations. When we write equations in matrix form, we align them into a neat structure that helps visualize and compute solutions easily. In matrix form, a system such as our ticket sales equations is expressed as:\[\begin{bmatrix}1 & 1 \11.15 & 5.95\end{bmatrix}\begin{bmatrix}x \y\end{bmatrix}=\begin{bmatrix}1200 \12756\end{bmatrix}\]The leftmost matrix is the coefficient matrix, capturing the relationship of variables \( x \) and \( y \) in our equations. The next matrix illustrates the variables themselves, and the rightmost vector represents the constants from the equations. Transforming equations into matrix form prepares them for advanced solving techniques like Cramer's Rule. This arrangement simplifies solving the system by allowing clear utilization of mathematical tools, offering convenience and clarity.