Problem 76 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $$\$ 12.50,$$ and the price for an adult ticket was $$\$ 16.00$$. The register confirms that $$\$ 5,075$$ was taken in. How many student tickets and adult tickets were sold?

Step-by-Step Solution

Answer

150 student tickets and 200 adult tickets were sold.

Step 1: Define Variables

To solve the problem, let's define variables for the number of tickets sold. Let \( s \) be the number of student tickets and \( a \) be the number of adult tickets.

Step 2: Set Up Equations

We have two pieces of information to convert into equations: the total number of tickets and the total amount of money collected. The first equation represents the total tickets: \( s + a = 350 \). The second equation represents the total sales: \( 12.50s + 16.00a = 5075 \).

Step 3: Solve the Equations Using Substitution

Solve the first equation for \( s \): \( s = 350 - a \). Substitute \( s = 350 - a \) into the second equation: \( 12.50(350 - a) + 16.00a = 5075 \).

Step 4: Simplify and Solve for Adult Tickets

Expand and simplify the equation: \( 4375 - 12.50a + 16.00a = 5075 \). Combine like terms to get \( 3.50a = 700 \). Solve for \( a \) by dividing both sides by \( 3.50 \): \( a = 200 \).

Step 5: Solve for Student Tickets

Substitute \( a = 200 \) back into \( s = 350 - a \) to find \( s \): \( s = 350 - 200 = 150 \).

Step 6: Verify the Solution

To ensure the solution is correct, check both original conditions: \( 12.50 \times 150 + 16.00 \times 200 = 1875 + 3200 = 5075 \), which matches the total sales. Also, \( 150 + 200 = 350 \), matching the total number of tickets.

Key Concepts

Solving Linear Systems

When faced with a scenario that involves finding multiple unknowns, like the number of student and adult tickets sold, we can use a system of linear equations to find a solution. Solving linear systems involves finding the values of variables that make all equations true at the same time. In our concert ticket problem, this means discovering how many student and adult tickets were sold.

To solve these systems, you typically set up a pair of equations based on the problem's information. Then, you can apply methods like substitution or elimination. Substitution involves solving one equation for a variable and then replacing it in the other equation, which helps simplify the process. In our case, we solved for one variable, substituted it into the second equation, allowing us to find precise values for both variables. This step-by-step approach helps ensure accuracy and clarity, making sure that all aspects match logically.

Variables and Equations

In algebra, variables represent unknown values that you're trying to determine. They're often symbolized by letters like \( s \) and \( a \), which stand for student and adult tickets in this exercise. Equations are mathematical statements that show the equality of two expressions. They are fundamental to representing real-world scenarios mathematically.

In this problem, we used two equations: one for the total number of tickets and another for the total sales. The variables allowed us to set up these equations:

• \( s + a = 350 \) for total tickets
• \( 12.50s + 16.00a = 5075 \) for total sales

These equations capture the relationship between the numbers of tickets and the financial outcome. By manipulating these equations using algebraic techniques, we derived the number of tickets sold, which are the variables' values.

Real-World Applications of Algebra

Algebra isn't just a theoretical subject; it's a practical tool that applies to everyday problems. Real-world applications range from personal financial planning to scientific research, but a simple yet effective example can be seen in business scenarios like event management.

In the concert ticket problem, algebra helps the manager understand how ticket sales translate into revenue. It assists in making informed decisions, like pricing and estimating future financial possibilities. Moreover, algebraic thinking offers a structured way to consider complexities, ensuring consistent logical reasoning aids in explaining and solving diverse situations.

With such tools, individuals can decipher data and understand various dynamics, thereby making smarter decisions. The concert example illustrates how algebra facilitates the management and prediction of financial outcomes, providing a basis for strategy and planning.