Problem 40 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The yearly salary is \(\$ 25,000\) plus \(\$ 75\) times the number of years of experience.

Step-by-Step Solution

Answer

The mathematical model is \( S = 25000 + 75x \).

Step 1: Understanding the Verbal Model

The verbal model describes a relationship where the yearly salary depends on two components: a fixed base amount of \( \\( 25,000 \) and a variable component that is \( \\) 75 \times \text{the number of years of experience} \).

Step 2: Identifying Variables

Let's define the variables. Let \( S \) represent the yearly salary, and \( x \) represent the number of years of experience. By doing this, we can easily translate the verbal information into a math equation.

Step 3: Translating to a Mathematical Model

With the variables identified, the verbal model can be written as an equation: \( S = 25000 + 75x \), where \( S \) is the yearly salary and \( x \) is the number of years of experience.

Key Concepts

Verbal Model

A verbal model is essentially a way to express relationships and scenarios using words rather than numbers or symbols. It's like a blueprint that helps set the stage before drafting the actual mathematical representation. In the context of our exercise, a verbal model is a sentence or a few sentences describing how different pieces of information connect. For example, the problem states that a yearly salary is comprised of a fixed base of \( \\(25,000 \) and an additional amount that increases with experience, \( \\)75 \) for each year. Such statements help think about what kind of mathematical elements you need to represent in a more numeric form later.

Using verbal models is crucial for understanding word problems, as they allow you to break down complex information into smaller, manageable parts. This makes it easier to find relationships and see how variables interact. As you become adept at creating verbal models, translating these into mathematical equations becomes more intuitive.

• Identify fixed numbers or constant amounts mentioned.
• Look for variables or changing quantities, usually described with phrases like "per each" or "times the number of."

Mathematical Equation

Once you have your verbal model, the next step is to translate it into a mathematical equation. An equation is a mathematical statement that shows the equality between two expressions. In our scenario, the equation is derived from the verbal model: the fixed salary of \( \\(25,000 \) plus \( \\)75 \) multiplied by the number of years of experience, represented by \( x \).

To form a correct mathematical equation:

• Identify what you are trying to find or solve for—in our case, the yearly salary.
• Turn descriptive terms into substitute variables (for example, let \( S \) be the salary and \( x \) be years of experience).
• Use mathematical symbols (+, *, =) to connect these variables and constants as described in the verbal model.

So, following our example, the mathematical equation would read as \( S = 25000 + 75x \). Each component of the verbal model finds a corresponding part in the equation, maintaining the equality and relationship described initially.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. They don't contain an equality sign like equations do. In our problem, while the equation \( S = 25000 + 75x \) defines the salary, each side of the equation contains algebraic expressions.

The right side of the equation \( 25000 + 75x \) is an algebraic expression that represents the total salary. It effectively brings together a constant and a variable term:

• Constant Term: This is a fixed value like \( 25000 \), which doesn't change, regardless of the value of experience.
• Variable Term: In this case, \( 75x \), where \( 75 \) is multiplied by the number of years of experience \( x \). This term represents the amount that varies depending on input values (experience).

Understanding these expressions is fundamental to forming equations, as they are the building blocks that summarize the variable parts of different scenarios. Recognizing which terms are constant and which are variable helps in manipulating and solving for unknowns within mathematical problems.