Problem 242 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Tickets for a baseball game are \(\$ 69\) for Main Level seats and \(\$ 39\) for Terrace Level seats. A group of sixteen friends went to the game and spent a total of \(\$ 804\) for the tickets. How many of Main Level and how many Terrace Level tickets did they buy?

Step-by-Step Solution

Answer

6 Main Level seats and 10 Terrace Level seats.

Step 1: Define the variables

Let the number of Main Level seats be denoted by \( x \) and the number of Terrace Level seats be denoted by \( y \).

Step 2: Set up the equations

Based on the information given, we can create the following system of equations:1. The total number of tickets equation: \( x + y = 16 \)2. The total cost of tickets equation: \( 69x + 39y = 804 \)

Step 3: Solve the first equation for one variable

Solve the first equation for \( x \): \[ x = 16 - y \]

Step 4: Substitute into the second equation

Substitute \( x = 16 - y \) into the second equation:\[ 69(16 - y) + 39y = 804 \]

Step 5: Simplify the equation

Distribute and simplify:\[ 1104 - 69y + 39y = 804 \]\[ 1104 - 30y = 804 \]

Step 6: Solve for \( y \)

Isolate \( y \) and solve:\[ 1104 - 804 = 30y \]\[ 300 = 30y \]\[ y = 10 \]

Step 7: Solve for \( x \)

Substitute \( y = 10 \) back into the first equation to find \( x \):\[ x + 10 = 16 \]\[ x = 6 \]

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent unknown values, and they show relationships between these variables using algebraic operations such as addition, subtraction, multiplication, and division.

An algebraic equation typically takes the form of one or more expressions set equal to each other. For example, in the context of our exercise, we have two equations:

\( x + y = 16 \) and \( 69x + 39y = 804 \).
The first equation represents the total number of tickets, while the second represents the total cost.
To solve these equations, we need to find the values of the variables that make both equations true simultaneously.

The Art of Problem-Solving with Systems of Equations

Solving a system of equations involves finding values for the variables that satisfy all equations in the system.

There are various methods to solve these systems, including substitution, elimination, and graphical methods.
Let's discuss how the substitution method was used in this exercise:

• Define the Variables: We defined \( x \) as the number of Main Level seats and \( y \) as the number of Terrace Level seats.
• Set up the Equations: We used the information given in the problem to set up our two key equations:
• Solve for One Variable: We solved the first equation for one variable, thus \( x = 16 - y \).
• Substitute into the Second Equation: We substituted the expression from the first equation into the second equation, resulting in a single variable equation: \( 69 ( 16 - y ) + 39y = 804 \).
• Simplify and Solve: This was followed by simplifying the new equation and solving for \( y = 10 \).
• Substitute Back: Lastly, we substituted \( y = 10 \) back into the first equation to find \( x = 6 \).

Understanding Variables

Variables are symbols used to represent unknown quantities in equations and mathematical expressions.
They allow us to write equations in a general form, making it easier to manipulate and solve them.

In the exercise, we defined two variables:

\(\text{ x } \) to represent the number of Main Level tickets, and

\(\text{ y } \) to represent the number of Terrace Level tickets.

This helped us to form our system of equations and find specific values for ticket numbers:

\(\text{x = 6, y = 10} \) .

By understanding and effectively using variables, you can tackle complex problems and break them down into manageable steps.