Problem 25 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. \(w\) reduced by 500

Step-by-Step Solution

Answer

The algebraic expression is \( w - 500 \).

Step 1: Identify the Key Information

The phrase given is "w reduced by 500". Here, 'reduced by' indicates subtraction, and 'w' is a variable representing an unknown quantity.

Step 2: Choose the Variable and Operation

The problem already specifies the variable 'w'. Since 'reduced by' suggests subtraction, we will use subtraction to represent this phrase algebraically.

Step 3: Construct the Algebraic Expression

To represent "w reduced by 500" in algebraic form, we subtract 500 from the variable w. The algebraic expression for this is \( w - 500 \).

Key Concepts

Variables in Algebra

In algebra, variables are symbols or letters that represent unknown values or quantities. These symbols allow us to create expressions and equations to model real-world situations. Variables are fundamental because they give us the ability to communicate mathematical ideas without knowing specific numbers. Here are a few points about the usefulness of variables:

• Variables can represent any number within a defined set, making them flexible for various applications.
• They help in generalizing mathematical rules and formulas.
• By using variables, complex relationships and operations can be expressed simply and concisely.

In the example exercise, "w" is chosen as a variable. It stands for an unspecified amount that can be reduced by 500. Choosing variables allows us to form the algebraic expression that will solve or describe the problem at hand. Therefore, understanding variables is crucial to navigating algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are used to represent real-world phenomena and solve mathematical problems. An algebraic expression does not contain an equality sign; it is not an equation, but rather a mathematical phrase that shows a calculation or value. Let's illustrate the components:

• A variable, like 'w', which signifies the quantity we want to identify or denote.
• Numbers (constants), which are fixed values in the expression.
• Operations, such as addition, subtraction, multiplication, and division.

In our given exercise, the algebraic expression is represented as \( w - 500 \). This expression instructs us to subtract 500 from the quantity symbolized by the variable \( w \). Crafting such expressions is pivotal as they lay the foundation for solving equations and developing mathematical models.

Subtraction in Algebra

Subtraction in algebra works much like subtraction in arithmetic; however, it involves variables alongside numbers. The phrase "reduced by" commonly signals subtraction. It is key to translating words into algebraic symbols. Here’s how subtraction is usually represented in algebra:

• The subtraction operation is indicated by the "-" sign.
• When subtracting a constant from a variable, it appears as \( \ ext{variable - constant} \). For example, \( w - 500 \) says 500 is subtracted from \( w \).

The order is crucial in subtraction. \( w - 500 \) is not the same as \( 500 - w \). The former indicates that an amount of 500 is taken away from what \( w \) represents, which is exactly the translation of the original phrase "w reduced by 500". Recognizing these subtle differences enhances algebraic understanding and prevents errors in problem-solving.