Problem 79 - Introductory Algebra for College Students

Question

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The model \(P=18 n+765\) describes the price of a Westie puppy, \(P, n\) years after \(1940,\) so I have to solve a linear equation to determine the puppy's price in 2009 .

Step-by-Step Solution

Answer

The statement 'I have to solve a linear equation to determine the puppy's price in 2009' does not make sense because the linear equation didn't need to be 'solved', instead it was 'used' to compute the price of a Westie puppy in 2009.

Step 1: Understand the linear equation

The given equation is \(P=18n+765\), where \(P\) is the price of the puppy, and \(n\) is the number of years after 1940.

Step 2: Calculate Value of \(n\)

We know that \(n\) is the number of years after 1940. In 2009, this would be \(n = 2009 - 1940 = 69\).

Step 3: Find the Price using the linear equation

Substitute \(n=69\) into the equation to get \(P = 18 * 69 + 765\). Calculate the result to get the price, \(P\).

Step 4: Analyze the given Statement

The statement was about 'solving' the equation to find the puppy's price in 2009. However, the procedure performed in step-3 was using the equation to 'find' the price, not to 'solve' the equation. An equation is 'solved' when we find the value(s) of its unknown(s), but here the year 2009 is known and not an unknown.

Key Concepts

Solving Equations

Linear equations are a fundamental concept in algebra that describe relationships between variables. In this case, the exercise provided the linear equation \(P=18n+765\), where \(P\) represents the price of a Westie puppy and \(n\) is the number of years after 1940. Solving an equation typically means finding the unknown values that satisfy it. However, in this exercise, we're given a specific year, 2009, and need to calculate \(n\) as it is based on known information.For linear equations, solving involves simple algebraic manipulations such as addition, subtraction, multiplication, or division to isolate the variable of interest. Here, instead of solving for an unknown, we directly substitute \(n = 69\) (since 2009 - 1940 = 69) to find \(P\), the puppy's price. This highlights an important distinction: solving for an unknown vs. computing a known variable's effect.

Algebraic Modeling

Algebraic modeling is used to represent real-world situations using mathematical equations. It involves identifying variables, setting up relationships between them, and forming equations to describe these relationships. In this exercise, the equation \(P = 18n + 765\) serves as a model to describe how the price of a puppy changes over time.

• \(P\): Represents the dependent variable, meaning it changes based on \(n\).
• \(n\): Represents the independent variable, calculated as years since 1940.

By using algebraic models, we can make mathematical predictions and understand trends. Here, the coefficient \(18\) suggests that for each additional year, the puppy's price increases by 18 units (dollars). The constant \(765\) represents the starting price in 1940. This model not only helps in calculating prices but also in understanding how the value evolves over time.

Mathematical Reasoning

Mathematical reasoning involves logically analyzing problems and drawing conclusions based on the given data. In this exercise, understanding and interpreting the context of the equation is crucial. The statement implies that solving is needed to find the price, but it's actually about finding the result for a specific known year. Mathematical reasoning allows us to assess whether a statement "makes sense." In this context, while you perform arithmetic operations to understand outputs from equations, it's important to differentiate solving for unknowns and plugging values into a known formula. By clearly defining the problem and understanding all variables and constants involved, we can make informed interpretations and avoid misconceptions. This skill is critical for tackling real-world problems and developing a solution-focused mindset.