Problem 42 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The difference between 500 and the number of people in a theater gives the number of unsold tickets.

Step-by-Step Solution

Answer

The mathematical model is \( 500 - x = y \).

Step 1: Define Variables

Let's define the variables we need to model the problem. Let \( x \) represent the number of people in the theater, and \( y \) represent the number of unsold tickets.

Step 2: Translate Verbal Model

The problem states that the difference between 500 and the number of people in the theater (\( x \)) is equal to the number of unsold tickets (\( y \)). In mathematical terms, this statement translates to the equation: \( 500 - x = y \).

Step 3: Finalize the Equation

Check the mathematical model to ensure it accurately represents the verbal statement. The equation \( 500 - x = y \) correctly represents that if you subtract the number of people in the theater from 500, you will get the number of unsold tickets.

Key Concepts

Variable Definition

When tackling any mathematical modeling problem, one of the first and most crucial steps is defining your variables. Variables are essentially placeholders or symbols that we use to represent unknown quantities in a problem. Think of them as simple stand-ins for the values we are trying to find or manipulate.

In this exercise, we needed to choose and define two variables to make sense of the problem. Variables can be literally anything you decide, but choosing labels that are intuitive and relevant to the context helps in understanding the problem better.

For example:

• We let \( x \) be the number of people in the theater.
• We let \( y \) be the number of unsold tickets.

The choice of letters does not matter mathematically, but using meaningful letters makes the equation easier to follow. Once you've defined your variables, you're ready to move on to the next steps: building and translating into equations.

Equation Building

Equation building is a step where the situation described in words turns into a mathematical statement. It's like building a bridge between the problem and the solution.

In our example, we are told that the difference between a total of 500 and the number of people in the theater equals the number of unsold tickets.

This directly corresponds to the equation:

• \( 500 - x = y \)

Here is how we constructed this equation:

• The number 500 is the total amount that represents either capacity, a maximum limit, or a fixed count.
• The subtraction from 500 indicates removal or difference from this total, pinpointing the essence of the problem.
• The result of subtracting the variable \( x \), symbolizing people in the theater, gives us \( y \), or unsold tickets.

This equation correctly encapsulates the relationship noted in the verbal statement. It’s a concise and exact representation of the given scenario.

Verbal to Algebraic Translation

This process is about translating ordinary language into the mathematical language of equations. Verbal to algebraic translation requires a careful reading and interpretation of the problem.

In our exercise, the verbal model states: "The difference between 500 and the number of people in a theater gives the number of unsold tickets." Each part of this statement corresponds to a part of an equation.

Let’s break it down:

• "The difference between 500 and the number of people in a theater" translates directly into the operation \( 500 - x \). The word 'difference' clues us into using subtraction.
• "Gives the number of unsold tickets" indicates that the result of the subtraction equals \( y \). This segment tells us what the subtraction operation results in.

Thus, we convert the problem statement into \( 500 - x = y \). This algebraic expression is clear and straightforward once words translate directly into mathematical symbols and operations.