Problem 33 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Operation Type

The phrase 'the product of' indicates a multiplication operation, and 'decreased by' indicates subtraction. We need to determine which numbers or variables to use in these operations.

Step 2: Define the Product Expression

The expression involves the product of the variable and a number. In this case, multiply the variable \(d\) by \(4\), resulting in the expression \(4d\).

Step 3: Formulate the Complete Expression

After obtaining the product \(4d\), the phrase 'decreased by 15' indicates we need to subtract 15 from this product. Thus, the expression becomes \(4d - 15\).

Key Concepts

Understanding Variables

Variables are symbols used in mathematics to represent unknown values or quantities. They are usually depicted by letters such as \(x\), \(y\), or \(d\) in our exercise. The beauty of using variables is their flexibility, as they allow you to represent a wide range of possible numbers. This is particularly helpful in algebraic expressions, where you need a placeholder for one or more numbers.
In our example, the variable \(d\) is used. It stands in for some unknown number that can change.

• Variables make expressions general rather than specific; they are not confined to one numerical value.
• They enable us to write equations that can solve multiple problems at once.

With this approach, you can use the same expression to find various answers based on different values of \(d\). This is the core reason why variables are foundational in algebra.

Exploring Multiplication

Multiplication is one of the basic operations in mathematics, often described as repeated addition. When we talk about multiplying a variable, it means that the variable is being counted repeatedly multiple times.
In the phrase "the product of \(d\) and \(4\)", multiplication plays a central role.
In this context, "product" means that you multiply the two specified numbers, the variable \(d\) and the constant \(4\). Thus, it becomes \(4d\).

• Multiplication can be visualized as making copies; \(4d\) means "four copies of \(d\)".
• When you multiply, the placement of the number can affect readability, such as \(4d\) instead of \(d4\).

By setting up your multiplication accurately, you help ensure the final expression correctly conveys the problem posed in the exercise.

The Role of Subtraction

Subtraction is another essential operation in mathematics, integral to forming complete algebraic expressions. It represents the process of taking one quantity away from another.
In our exercise, subtraction comes into play with the words "decreased by 15."
When you see this phrase, it means you need to subtract \(15\) from the expression you have already created, which here is \(4d\). Hence, it transforms into \(4d - 15\).

• Subtraction takes away from the total value of the previous operation, which in this case is the multiplication.
• The order matters; ensure subtraction is performed after multiplication for accurate results.

Understanding the sequence of operations is critical. It helps you to form expressions where subtraction changes the total valence appropriately, reflecting real-world conditions described by the problem.