Problem 43 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. Each test score was increased by 15 points to give a new adjusted test score.

Step-by-Step Solution

Answer

\( y = x + 15 \)

Step 1: Define the Variables

Start by defining a variable for the original test score. Let's use \( x \) to represent the original test score. The adjusted test score will be represented by \( y \).

Step 2: Formulate the Verbal Model

From the problem, we know that each test score was increased by 15 points. This means the new adjusted test score \( y \) is equal to the original test score \( x \) plus 15 points.

Step 3: Translate to a Mathematical Model

Using the verbal model, we can write the equation for the adjusted test score as \( y = x + 15 \). This captures the idea that each score increases by 15.

Key Concepts

Variables in Algebra

In algebra, variables play a crucial role in forming mathematical models and solving problems. A variable is essentially a symbol, often a letter like \( x \) or \( y \), that represents a number or a set of numbers. In our original exercise, we defined variables to represent test scores: \( x \) for the original test score, and \( y \) for the adjusted score.

Understanding variables is important because they allow us to create expressions and equations that describe various scenarios. By assigning a variable to a particular unknown aspect of a problem, we can then manipulate these expressions according to algebraic rules to find solutions.

• Variables can be independent or dependent. An independent variable is one that we can freely change, whereas a dependent variable depends on the value of the independent variables.
• In equations, variables enable us to express relationships between quantities.
• They also help in generalizing problems so that we can apply the same principles to a wide range of situations.

Using variables like \( x \) and \( y \) makes it easier to set up equations that we can solve for unknown values, assisting us in modeling real-life problems mathematically.

Linear Equations

Linear equations are a type of algebraic equation that represent straight lines when graphed on a coordinate plane. In simple terms, they express a linear relationship between two variables, typically in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

In the exercise given, the linear equation \( y = x + 15 \) arises by increasing the test score by 15 points. This represents a situation where the adjusted score (\( y \)) is directly related to the original score (\( x \)) plus a constant increase. Linear equations are straightforward because:

• They show a constant rate of change—meaning the increase or decrease in \( y \) is always the same for a change in \( x \).
• The constant term (in this case, 15) shifts the line vertically without altering its slope.
• Such equations are used to model many real-life situations where relationships between quantities are consistent.

Recognizing and using linear equations enables efficient problem-solving, translating verbal conditions into mathematical terms that are easier to analyze and graphically represent.

Problem-Solving in Algebra

When tackling problems in algebra, the key is translating word problems into mathematical representations. This process relies on identifying variables and forming equations that reflect the given situation. For the test score problem, the steps involved:

1. **Identifying the variables:** We determined which quantities in the problem are unknowns we need to solve for. Here, the original test score and the adjusted score were the variables.

2. **Constructing the model:** We articulated the situation as a mathematical model: "Each test score was increased by 15 points," translating this into our linear equation, \( y = x + 15 \).

3. **Solving the equation:** Though the problem didn’t require solving for specific values, typically, the next step would involve using algebraic methods to find values for \( x \) and \( y \).

• Solving involves substituting known values, simplifying expressions, and checking solutions.
• Algebraic problems often carry a logical sequence of steps: define, translate, and manipulate.

By following these steps, algebra becomes a powerful toolkit that allows us to systematically and logically work through problems, leading to more accurate and reliable solutions.