Problem 62 - 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. Three bands performed at a concert venue. The first band charged \(\$ 15\) per ticket, the second band charged \(\$ 45\) per ticket, and the final band charged \(\$ 22\) per ticket. There were 510 tickets sold, for a total of \(\$ 12,700\). If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

Step-by-Step Solution

Answer

150 tickets for the first band, 110 for the second, 250 for the third.

Step 1: Define the Variables

Let's define the variables for the tickets sold by each band. Let \( x \) be the number of tickets sold by the first band, \( y \) be the number of tickets sold by the second band, and \( z \) be the number of tickets sold by the final band.

Step 2: Set Up the Equations

Using the given information, we can set up the following equations:1. The total number of tickets sold is 510: \[ x + y + z = 510 \]2. The total revenue generated is $12,700: \[ 15x + 45y + 22z = 12700 \]3. The first band had 40 more audience members than the second band: \[ x = y + 40 \]

Step 3: Substitute and Simplify

We can substitute \( x = y + 40 \) from the third equation into the first two equations to reduce the number of variables:1. Substitute into the first equation: \[ (y + 40) + y + z = 510 \] Simplifying gives: \[ 2y + z = 470 \]2. Substitute into the second equation: \[ 15(y + 40) + 45y + 22z = 12700 \] Simplifying gives: \[ 60y + 22z = 12100 \]

Step 4: Solve Simultaneously Using Cramer's Rule

Our simplified system of equations is:1. \[ 2y + z = 470 \]2. \[ 60y + 22z = 12100 \]Using Cramer's Rule, the matrix form is:\[ A = \begin{bmatrix} 2 & 1 \ 60 & 22 \end{bmatrix}, \quad B = \begin{bmatrix} 470 \ 12100 \end{bmatrix} \]First, we find the determinant of matrix \( A \):\[ \text{det}(A) = (2)(22) - (1)(60) = 44 - 60 = -16 \]Next, we find the determinants of matrices \( A_y \) and \( A_z \). Replace the column of \( y \) with \( B \) in \( A \) to form \( A_y \), and similarly for \( A_z \):\[ A_y = \begin{bmatrix} 470 & 1 \ 12100 & 22 \end{bmatrix}, \quad A_z = \begin{bmatrix} 2 & 470 \ 60 & 12100 \end{bmatrix} \]Calculate \( \text{det}(A_y) \):\[ \text{det}(A_y) = (470)(22) - (1)(12100) = 10340 - 12100 = -1760 \]Calculate \( \text{det}(A_z) \):\[ \text{det}(A_z) = (2)(12100) - (470)(60) = 24200 - 28200 = -4000 \]Now apply Cramer's Rule:\[ y = \frac{-1760}{-16} = 110 \]\[ z = \frac{-4000}{-16} = 250 \]Finally, use \( x = y + 40 \):\[ x = 110 + 40 = 150 \]

Step 5: Conclusion

Thus, the number of tickets sold by the first band is 150, the second band is 110, and the final band is 250.

Key Concepts

Systems of Linear Equations

In mathematics, systems of linear equations are collections of two or more linear equations involving the same set of variables. These equations represent lines, planes, or hyperplanes, depending on the number of variables involved. By solving these systems, we aim to find the value of each variable that satisfies all the equations simultaneously, meaning these values represent a point where all equations intersect. In practical problems, like the Ticket Sales Problem described, these can model real-world scenarios allowing for relationship mapping between different quantities.

To formulate a system of equations, we often express constraints or relationships between various elements using equalities. For example, in the ticket sales problem, one equation accounted for the total tickets sold, another for the total amount of money collected, and a third for the relationship between the two bands.

• Equation 1: Sum of the tickets sold for all bands.
• Equation 2: The total revenue of all tickets sold.
• Equation 3: Difference in audience members between the first and second band.

Each equation introduces a new constraint or relationship allowing us to solve for the variables involved.

Matrix Determinant

The determinant is a specific number calculated from a square matrix, which can offer insights into the properties and characteristics of the matrix itself. In particular, it provides crucial information in the context of solving systems of linear equations using techniques like Cramer's Rule. A non-zero determinant indicates that the set of equations has a unique solution, whereas a determinant of zero signifies either no solutions or infinitely many solutions.

In the Ticket Sales Problem, we dealt with a 2x2 coefficient matrix, denoted as \( A \), which was derived from the simplified system of equations. The determinant of this matrix \( A \) is computed by the formula:
\[ \text{det}(A) = ad - bc \]
where \( a \), \( b \), \( c \), and \( d \) are elements of the matrix.

In our example:

• The matrix \( A \) was \( \begin{bmatrix} 2 & 1 \ 60 & 22 \end{bmatrix} \).
• The determinant \( \text{det}(A) \) was \( (-16) \), guaranteeing a unique solution exists.

The determinant aids in solving linear equations where methods like substitution or elimination are less efficient or impossible.

Ticket Sales Problem

The Ticket Sales Problem is a classic example of modeling real-world scenarios using algebraic methods. Here, we have three bands, each charging a different price per ticket, selling a total of 510 tickets combined, resulting in specific revenue. Problems like this typically require setting up equations based on given information and constraints, then solving the system for unknown variables.

Through this exercise, variables \( x \), \( y \), and \( z \) were used to represent the tickets sold by each of the bands. Using the system of equations we formulated, we translated real-world data into mathematical language:

• One equation signified the sum of all tickets sold equaling 510.
• Another represented the collective revenue of $12,700 from all bands.
• A final one expressed that the first band sold 40 more tickets than the second.

These equations were simplified and solved using mathematical techniques such as substitution and Cramer's Rule, arriving at the number of tickets each band sold. This structured approach mirrors how algebra is systematically used to tackle problem-solving scenarios.

Algebra Problem Solving

Algebra problem solving involves using algebra to find unknown values from given information. It’s a fundamental skill in mathematics that applies to various fields like engineering, physics, economics, and beyond. Key techniques include manipulating equations, substituting given values, and applying rules like Cramer's Rule to arrive at solutions.

In exercises like the Ticket Sales Problem, solving involves:

• Defining variables to represent unknown quantities.
• Creating equations to model the problem situation.
• Using algebraic techniques such as substitution and elimination to simplify solving.

Cramer's Rule specifically provides an elegant way to solve certain systems of equations when expressed in matrix form. With the determinant non-zero, we can effectively find each variable's value via matrix manipulation.

Successful algebra problem solving hinges on clearly stating the problem, understanding relationships within given data, and methodically applying mathematical principles to rearrange and solve equations.