Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Business \(A\) minor league baseball team had a total attendance one evening of \(1175 .\) The tickets for adults and children sold for \(\$ 15\) and \(\$ 12,\) respectively. The ticket revenue that night was \(\$ 16,275\) (a) Create a system of linear equations to find the numbers of adults \(A\) and children \(C\) at the game. (b) Solve your system of equations by elimination or by substitution. Explain your choice. (c) Use the intersect feature or the zoom and trace features of a graphing utility to solve your system.

Step-by-Step Solution

Answer

725 adults and 450 children attended the game.

Step 1: Identify the Given Variables

The numbers of adults and children who attended the game are represented as A and C respectively. The vital pieces of information are that the total audience equals 1175 and the total revenue made from ticket sales is $16,275; adult tickets sell for $15 and children's tickets for $12.

Step 2: Form the System of Linear Equations

Create two equations based on the available information. The first equation is formed by the total number of people who attended the game: \(A + C = 1175\). The second equation is formed using the total ticket revenue: \(15A + 12C = 16275\)

Step 3: Choose a Method to Solve the System of Equations

Choose the method of elimination to solve this system, as it allows you to eliminate variables more effectively. To make it easier, multiply the first equation by 12, giving us \(12A + 12C = 14100\). Subtraction of this equation from the second one gives you \(3A = 2175\)

Step 4: Solve for A

Divide by 3 to find A: \(A = 2175 / 3 = 725\) adults attended the game.

Step 5: Solve for C

Substitute \(A = 725\) into the first equation \(A + C = 1175\), and solve for C, which gives you \(C = 1175 - 725 = 450\) children attended the game.

Step 6: Verification Using a Graphing Utility

Use a graphing utility to plot the equations of the system: \(A + C = 1175\) and \(15A + 12C = 16275\) . The point of intersection of these lines should correspond to the solutions for A and C found earlier.

Key Concepts

Solving Linear Systems

The process of solving linear systems involves finding values for variables that satisfy all equations within the system simultaneously. In our textbook example, we are given a real-world scenario where we need to calculate the number of adult and child tickets sold for a baseball game. The system consists of two equations that reflect the total number of attendees and the total revenue earned.

The goal is to find specific values for the variables representing adults, denoted by A, and children, denoted by C, which make both equations true at the same time. Solving such systems can utilize various methods, each with its own advantages and application scenarios. Understanding how to solve these systems is invaluable as they frequently represent realistic situations like the one presented in the exercise.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by removing one variable to find the value of another. This method often simplifies the process when equations can be easily manipulated to cancel out one variable. In the provided exercises, students are encouraged to use elimination method by modifying the coefficients of the variables to be equal. By making the coefficients of C the same in both equations, C can be efficiently eliminated when one equation is subtracted from the other.

This leaves an equation with only one variable, A, which can then be solved directly. This technique is especially powerful when used in combination with multiplication or division to adjust the coefficients as demonstrated in the step by step solution. By understanding the elimination method, students can tackle a wide range of linear systems with confidence.

Substitution Method

Alternatively, the substitution method is another strategy to solve a system of linear equations, which can be particularly useful when one equation is easy to solve for one variable. This involves rearranging one of the equations to express one variable in terms of the other and then substituting this expression into the other equation.

For instance, if we had chosen the substitution method for our exercise, we might solve the first equation for A, yielding A = 1175 - C, and plug this into the second equation. This would give us an equation in just one variable, C, which we could solve directly. The substitution method can be highly effective for systems where the elimination method might require more complex algebraic manipulations.

Graphical Solution

A graphical solution to a system of linear equations involves plotting each equation as a line on the coordinate plane and identifying the point(s) where the lines intersect. This intersection represents the solution to the system, corresponding to the values of the variables that satisfy all equations.

In educational settings, a graphical approach not only helps to visually demonstrate the concept of solving linear systems but also serves as a verification tool. In our context, students can use graphing calculators or software to plot the two equations from the game attendance scenario. By using the intersect feature or the zoom and trace features of a graphing utility, they can confirm the mathematically derived answers. The graphical approach is particularly engaging for visual learners and can help demystify abstract algebraic concepts.

Application in Real-world Problems

Systems of linear equations are not only academic exercises; they have many applications in real-world problems. The ticket sales example provided in the exercise is a practical manifestation of how linear systems can model economic situations, such as predicting revenues, understanding consumer behavior, or managing resources.

In diverse fields such as business, engineering, science, and even social sciences, linear systems serve as fundamental tools for making informed decisions based on multiple constraints and variables. Emphasizing the real-world application of these mathematical concepts can inspire students to appreciate the value of their academic pursuits and potentially spark interest in STEM careers.