Problem 76 - College Algebra

Question

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 confi ms 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 the Variables

First, let's define our variables. Let \( s \) represent the number of student tickets sold, and \( a \) represent the number of adult tickets sold.

Step 2: Set Up the Equations

We have two key pieces of information. The total number of tickets sold was 350, and the total revenue was $5,075. This gives us the following equations:1. \( s + a = 350 \)2. \( 12.5s + 16a = 5075 \)

Step 3: Solve the System of Equations - Elimination Method

To solve these equations, we can use the elimination method. Let's multiply the first equation by 12.5 to align the student ticket term: \( 12.5s + 12.5a = 4375 \)We'll have:\( 12.5s + 16a = 5075 \)\( 12.5s + 12.5a = 4375 \)

Step 4: Subtract the Equations to Eliminate s

Subtract the second equation from the first:\( (12.5s + 16a) - (12.5s + 12.5a) = 5075 - 4375 \)This simplifies to:\( 3.5a = 700 \)

Step 5: Solve for a

Now, solve for \( a \):\( a = \frac{700}{3.5} = 200 \).So, 200 adult tickets were sold.

Step 6: Solve for s

Substitute \( a = 200 \) back into the first equation to find \( s \):\( s + 200 = 350 \)\( s = 350 - 200 \)\( s = 150 \).So, 150 student tickets were sold.

Key Concepts

Elimination Method

The elimination method is a popular technique used in solving systems of linear equations, particularly when you have two equations with two variables. This approach is called "elimination" because you add or subtract the equations to eliminate one of the variables. This simplifies the problem to a single-variable equation, making it easier to solve.

• Start by aligning the equations such that one of the variables has the same coefficient in both equations.
• You can achieve this by multiplying one or both equations by constants if necessary.
• Once coefficients are aligned, add or subtract the equations to eliminate one variable.
• After elimination, solve the resulting single-variable equation.

By focusing on reducing the number of variables, the elimination method makes it more straightforward to find a solution to complex equations. Once one variable is eliminated, you can easily find the value of the remaining variable through simple algebra.

Variables

In the context of linear equations, variables are symbols used to represent unknown values that we need to solve for. They can stand for anything, such as quantities or measurements, and are often represented by letters like \( x \), \( y \), \( s \), or \( a \).

• In our exercise, the variables are \( s \) and \( a \), representing the number of student and adult tickets sold, respectively.
• By using variables, you can create equations that model real-world problems, allowing them to be solved mathematically.
• Defining your variables clearly is a crucial initial step in solving any system of equations.

Variables play a significant role in translating word problems into mathematical models, thus enabling problem-solving through algebraic methods.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when plotted on a coordinate plane. They are characterized by variables raised to the power of one (no exponents beyond one). Each linear equation can be written in the standard form: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

• Our two linear equations here are: \( s + a = 350 \) and \( 12.5s + 16a = 5075 \).
• These equations come from solving a real-world problem involving ticket sales and total revenue.
• Linear equations can have one solution, no solution, or infinitely many solutions, depending on the intersection of the lines they represent.

Understanding linear equations is essential for solving systems since it helps to visualize the relationship between variables on a graph.

Solving Equations

Solving equations involves finding the values for variables that satisfy the equations. For a system of linear equations with two variables, the goal is to find a pair of values, one for each variable, that makes both equations true at the same time.

• Start by choosing a method—elimination, substitution, or graphing.
• Use the chosen method to simplify the equations into a one-variable equation when dealing with two-variable systems.
• Solve the simplified equation for one variable, then substitute back into one of the original equations to find the other variable.

Solving systems can often require patience and careful arithmetic, but understanding how each step brings you closer to the solution makes the process logical and methodical.