Problem 56 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. the product of 4 and \(d,\) decreased by 15 b. the product of 4 and \(d\) decreased by 15

Step-by-Step Solution

Answer

The algebraic expression is \(4d - 15\).

Step 1: Identify the Components

First, let's identify the key information in the phrase. We're given the product of 4 and \(d\), and this result is decreased by 15.

Step 2: Formulate Algebraic Expression for Product

The product of 4 and \(d\) can be expressed as \(4 \times d\) or simply \(4d\).

Step 3: Incorporate the Decrease

To account for the phrase 'decreased by 15,' we subtract 15 from \(4d\). The algebraic expression becomes \(4d - 15\).

Step 4: Verify Consistency

Ensure that the expression encapsulates the original phrase. 'The product of 4 and \(d\)' is represented by \(4d\), and 'decreased by 15' is represented by subtracting 15, giving a complete expression of \(4d - 15\).

Key Concepts

Understanding Variables

In algebra, variables are essential components. They represent unknown values or quantities and are often denoted by letters such as \(x\), \(y\), or \(d\). Variables stand in place of numbers that can change or numbers that are not yet known. For example, in the phrase "the product of 4 and \(d\), decreased by 15," \(d\) is a variable that can take any number based on the context of the problem. This use of variables allows for a broader exploration of algebraic concepts without being tied to specific numbers. Variables provide the flexibility needed to discover relationships and patterns in mathematics.

Translation of Phrases into Algebraic Expressions

Translating phrases into algebraic expressions involves converting words into mathematical symbols in a logical manner. This is a fundamental skill in algebra. Let's look at the exercise: "the product of 4 and \(d\), decreased by 15." Here, 'the product of 4 and \(d\)' means multiplying 4 by \(d\), which is written as \(4d\). 'Decreased by 15' indicates a subtraction of 15 from the product. Bringing these elements together, you get the algebraic expression \(4d - 15\). This conversion is crucial as it forms the basis for solving algebraic problems, allowing these phrases to be manipulated and understood mathematically.

Mathematical Operations

Mathematical operations are the actions we perform with numbers and variables, like addition, subtraction, multiplication, and division. In our example, two operations are involved:

• Multiplication: The phrase "the product of 4 and \(d\)" translates to the multiplication of 4 by \(d\), or \(4d\).
• Subtraction: The phrase "decreased by 15" means we subtract 15 from \(4d\). This results in the expression \(4d - 15\).

Understanding these operations is key to forming and solving equations. Each operation follows specific rules that lead us to the correct mathematical expression.

Algebra Concepts

Algebra is the area of mathematics that uses symbols and variables to express relationships, patterns, and changes. At its core, algebra builds on basic arithmetic, but it introduces concepts that allow us to generalize and find solutions to a broad range of problems.
In our exercise, the expression \(4d - 15\) encapsulates several algebra concepts:

• Using variables to represent unknown quantities.
• Employing operations like multiplication and subtraction to transform these variables.
• Recognizing the structure of algebraic expressions and how to manipulate them.

Working with algebra expressions and equations develops critical thinking, problem-solving skills, and helps in understanding how mathematical concepts relate to each other and to real-world situations.