Problem 244 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Tickets for the community fair cost \(\$ 12\) for adults and \(\$ 5\) dollars for children. On the first day of the fair, 312 tickets were sold for a total of \(\$ 2,204 .\) How many adult tickets and how many child tickets were sold?

Step-by-Step Solution

Answer

92 adult tickets and 220 child tickets were sold.

Step 1: Define Variables

Let \(x\) represent the number of adult tickets sold and \(y\) represent the number of child tickets sold.

Step 2: Set Up Equations

Based on the problem, set up the two equations:1. The total number of tickets: \(x + y = 312\)2. The total revenue: \(12x + 5y = 2204\)

Step 3: Solve the System of Equations

First, use the first equation to express \(y\) in terms of \(x\): \(y = 312 - x\). Then substitute this into the second equation:\(12x + 5(312 - x) = 2204\)

Step 4: Simplify the Equation

Expand and simplify the equation:\(12x + 1560 - 5x = 2204\)Combine like terms:\(7x + 1560 = 2204\)

Step 5: Isolate \(x\)

Subtract 1560 from both sides:\(7x = 644\)Divide both sides by 7:\(x = 92\)

Step 6: Find \(y\)

Substitute \(x\) back into the equation \(y = 312 - x\):\(y = 312 - 92 = 220\)

Step 7: Verify the Solution

Check by substituting \(x\) and \(y\) into the total revenue equation: \(12(92) + 5(220) = 1104 + 1100 = 2204\).Both the total number of tickets and the revenue match the given problem, so the solution is correct.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols represent numbers and are used to express mathematical relationships. In the given exercise, we use algebra to form equations that describe the problem.

When dealing with word problems in algebra, the first step is to translate the problem into mathematical expressions. This involves:

• Defining variables to represent unknown quantities.
• Setting up equations based on the relationships described in the problem.

In our example exercise, we defined two variables:

• Letting \( x \) be the number of adult tickets sold.
• Letting \( y \) be the number of child tickets sold.

This step is crucial as it transforms the word problem into a form we can solve using mathematical operations.

Linear Equations

Linear equations are equations of the first degree, meaning they have the highest exponent of the variable as one. In simpler terms, they form a straight line when graphed. In the exercise provided, we encounter two linear equations:

1. \( x + y = 312 \)
2. \( 12x + 5y = 2204 \)
These equations arise from the problem statement. The first equation represents the total number of tickets sold, while the second represents the total revenue.

To solve a system of linear equations, methods such as substitution or elimination are used. Here, we used substitution:

• First, we solved the first equation for \( y \): \( y = 312 - x \).
• Then, we substituted this value into the second equation, resulting in a single equation with one variable:
• \( 12x + 5(312 - x) = 2204 \)

This simplifies the problem, allowing us to find the values of \( x \) and \( y \) more easily.

Problem-Solving

Problem-solving in mathematics involves identifying the problem, choosing a strategy to solve it, and executing the plan. Follow these steps to tackle similar exercises:

1. **Understand the problem:** Read the problem carefully and identify what is being asked. In this case, we needed to find the number of adult and child tickets sold.
2. **Formulate the plan:** Choose suitable variables and set up equations that model the problem's conditions. For our exercise, we defined \( x \) and \( y \), and wrote equations reflecting the total tickets and revenue.
3. **Execute the plan:** Use algebraic methods to solve the equations. Here, we used substitution and simplification to find the values of \( x \) and \( y \).4. **Verify the solution:** Check your results by substituting them back into the original equations. In our example, we verified that \( 92 \) adult tickets and \( 220 \) child tickets add up to both the total number of tickets and the total revenue.

By following these steps methodically, complex problems become easier to solve.