Problem 185 - Elementary Algebra

Question

The box office sold 360 tickets to a concert at the college. The total receipts were \(\$ 4170\). General admission tickets cost \(\$ 15\) and student tickets cost \(\$ 10\). How many of each kind of ticket was sold?

Step-by-Step Solution

Answer

114 general admission tickets and 246 student tickets were sold.

Step 1: Define variables

Let’s define the variables. Let: - \(x\) be the number of general admission tickets sold. - \(y\) be the number of student tickets sold.

Step 2: Set up the equations

We have two pieces of information: 1. The total number of tickets sold is 360. 2. The total receipts are \(\$4170\). From these, we can set up two equations: - Equation 1: \(x + y = 360\) - Equation 2: \(15x + 10y = 4170\)

Step 3: Solve the system of equations

First, solve Equation 1 for one of the variables. Let's solve for \(y\): \[y = 360 - x\] Next, substitute \(y\) in Equation 2: \[15x + 10(360 - x) = 4170\]

Step 4: Simplify and solve for x

Simplify the equation: \[15x + 3600 - 10x = 4170\] Combine like terms: \[5x + 3600 = 4170\] Subtract 3600 from both sides: \[5x = 570\] Divide by 5: \[x = 114\]

Step 5: Solve for y

Substitute \(x\) back into the expression for \(y\): \[y = 360 - x\] \[y = 360 - 114\] \[y = 246\]

Step 6: State the final answer

The number of general admission tickets sold is \(114\) and the number of student tickets sold is \(246\).

Key Concepts

Variable Definition

In algebra, we often use variables to represent unknown quantities. Variables are symbols, usually letters, that stand in for values we want to find.

For our problem, we need to determine how many general admission tickets and student tickets were sold.
Let's define our variables:

• Let x be the number of general admission tickets sold.
• Let y be the number of student tickets sold.
  
  By defining these variables, we transform the problem into one that can be solved using algebra.

Linear Equations

Linear equations are equations that graph as straight lines. They can be written in the form: ax + by = c where a, b, and c are constants.

In our exercise, we created two linear equations based on the given information:

x + y = 360 (total tickets sold)
1. 15x + 10y = 4170 (total receipts).
   
   These equations help us relate the number of each type of ticket sold with the total sales and total revenue.

Substitution Method

One of the techniques to solve a system of equations is the substitution method. This involves solving one equation for one variable and then substituting that expression into another equation.

For our problem, we solved x + y = 360 for y, giving us:

y = 360 - x
• We then substituted this expression into the second equation: 15x + 10(360 - x) = 4170.
  
  This replacement allows us to solve for one variable first, and then find the other.

Simultaneous Equations

Simultaneous equations are sets of equations that are solved together because they share the same variables.

In our case, we have the simultaneous equations:

• x + y = 360
• 15x + 10y = 4170
  
  To find the values of x and y, we combine the equations using methods like substitution, which ensures that the solutions satisfy both equations at the same time.
  
  This process helps us find the number of general admission and student tickets sold simultaneously.

Algebraic Problem Solving

Algebraic problem solving involves using mathematical concepts and techniques to find unknown variables. In our exercise, we used several algebraic steps:

• Defining variables
• Setting up equations
• Using substitution to simplify the system
• Solving the simplified equation
• Verifying the solution
  
  This step-by-step approach ensures that each part of the problem is systematically addressed, leading to clear and accurate solutions.