Problem 9 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. a. 7, times, the age of a dog in years, gives, the dog's equivalent human age. b. The take-home pay, will be, \(\$ 2,500\), minus, any deductions.

Step-by-Step Solution

Answer

(a) 7d; (b) 2500 - x

Step 1: Identify Variables

For the first part (a), let the dog's age be represented by the variable \(d\). For the second part (b), let the deductions be represented by \(x\).

Step 2: Create Verbal to Algebraic Expressions

For part (a), the phrase '7 times the age of a dog in years' implies multiplication, translating to \(7 \times d\). For part (b), the phrase 'will be \$2,500 minus any deductions' translates to \(2500 - x\).

Step 3: Formulate the Mathematical Models

For part (a), the equivalent human age is \(7d\). For part (b), the equation for take-home pay is \(2500 - x\).

Step 4: Review and Validate

Check that the translations align with the descriptions. In part (a), '7 times the age' is correctly multiplying the variable by 7, and in part (b), 'minus any deductions' accounts for subtracting \(x\) from 2500.

Key Concepts

Algebraic Expressions

In mathematics, algebraic expressions are used to represent numbers and operations on these numbers in a concise manner. This type of expression involves variables, constants, and mathematical operations. They are like phrases in a language, using variables (letters representing unknowns or quantities) and constants (fixed numbers) to convey a specific mathematical idea or relationship.
For example, in the exercise given, "7 times the age of a dog in years" is represented as the algebraic expression \(7d\). Here, \(7\) is the constant, and \(d\) is the variable representing the dog's age. Similarly, \(2500 - x\) is another expression where \(2500\) is a constant indicating the take-home pay without deductions, and \(x\) is the variable for deductions.
Algebraic expressions form the bedrock of many complex mathematical problems and are essential for modeling real-world situations in a form that can be manipulated and solved mathematically.

Variables in Mathematics

Variables are an integral part of algebra, serving as placeholders for unknown or variable quantities. They are typically denoted by letters such as \(x\), \(y\), or \(d\), as seen in our exercise. These letters can represent numbers whose values can change based on the context or conditions of a problem.
Variables make it easier to generalize mathematical rules and solutions. For instance, if you know that a dog's age in dog years converts to human years by multiplying by 7, the variable \(d\) in \(7d\) can represent any dog’s age, allowing flexibility and a broad application of this expression.
In practical problems, such as calculating the take-home pay from a salary, a variable like \(x\) represents deductions, which can differ from one paycheck to another. This adaptability is vital for modeling and solving problems across various disciplines, including finance, engineering, and science.

Verbal to Algebraic Translation

Translating verbal statements into algebraic expressions or equations is a vital skill in mathematical modeling. It involves interpreting the language of math into symbols and numbers that express the same idea more succinctly.
First, identify the variable by recognizing what quantity is changing or needs to be found. For example, in the sentence "7 times the age of a dog in years," the age is the unknown, so we represent it with a variable, such as \(d\).
Next, decipher the operations described verbally. The word "times" suggests multiplication, leading to the expression \(7d\). Similarly, in "will be $2,500 minus any deductions," the phrase "minus any deductions" translates to subtraction, resulting in \(2500 - x\).

• "Times" signals multiplication.
• "Minus" indicates subtraction.
• "Plus" would suggest addition.

Mastering verbal to algebraic translation allows for efficient problem-solving and better comprehension of mathematical concepts, bridging word problems with symbolic mathematics. This skill helps to transform detailed, lengthy descriptions into manageable mathematical models.