Problem 68 - Algebra for College Students

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. A number subtracted from 75

Step-by-Step Solution

Answer

The algebraic expression is \(75 - n\).

Step 1: Identify the phrase

The phrase given in the problem is "A number subtracted from 75." We need to translate this into an algebraic expression.

Step 2: Determine the Operation

The phrase "subtracted from" indicates a subtraction operation. In algebra, subtraction means that one value will be subtracted from another.

Step 3: Identify the Unknown

The unknown number in this problem is represented by the variable \(n\).

Step 4: Construct the Algebraic Expression

To translate the phrase "a number subtracted from 75," we write \(75 - n\). Here, \(n\) is being subtracted from 75.

Key Concepts

Subtraction in Algebra

Subtraction in algebra involves taking one value away from another. This basic arithmetic operation translates directly into algebra, where numbers can be replaced by variables.

In this exercise, the phrase given uses the words "subtracted from," which is a clear indicator to perform subtraction. It's crucial to note the order in which subtraction occurs, as it directly affects the outcome of the expression. For instance, "a number subtracted from 75" becomes \(75 - n\), not \(n - 75\). This is because in the phrase, 75 is what remains after the subtraction takes place, indicating 75 is the starting point.

Remember, subtraction is not commutative, which means \(a - b\) does not yield the same result as \(b - a\). So it's important to pay close attention to the phrasing when translating to ensure accuracy. Keep these points in mind whenever you come across subtraction operations in mathematical expressions.

Translating Phrases into Expressions

Translating phrases into algebraic expressions is a foundational skill in algebra that involves interpreting verbal statements or phrases and converting them into mathematical language. This process allows one to express complicated ideas succinctly using symbols and variables.

Consider the phrase "a number subtracted from 75." When translating, start by identifying key terms and operations:

• "A number": This indicates an unknown value that we can represent with a variable. In this problem, the variable is \(n\).
• "Subtracted from": This specifies the operation, which in this case is subtraction. The order is important here: the number comes after 75 in the expression.
• "75": This is a known number or constant that forms part of the expression.

By organizing these components, we form the expression \(75 - n\). This expression succinctly captures the verbal phrase, allowing for direct computation once \(n\) is known. Practicing this skill can improve your algebraic thinking and ability to solve diverse problems.

Use of Variables in Algebra

Variables play a crucial role in algebra, as they provide a way to represent unknown values within expressions and equations. Using a letter like \(n\) to signify an unknown number is a powerful way of generalizing problems so they can be solved regardless of specific numbers.

In the current exercise, \(n\) is used to denote the unknown quantity being subtracted from 75. This use of variables allows us to easily manipulate and transform algebraic expressions:

• They allow us to set up equations that can then be solved to find the unknown value.
• Variables provide flexibility, making it possible to use the same expression in varied contexts, simply by changing the value \(n\) takes.
• They help in expressing patterns and relationships between numbers graphically and numerically.

By mastering the use of variables, you can unlock more complex algebraic manipulations and develop a deeper understanding of how algebra models real-world situations.