Problem 3 - College Algebra

Question

In Exercises \(1-14\), let \(x\) represent the number. Write each English phrase as an algebraic expression. A number subtracted from 20

Step-by-Step Solution

Answer

The algebraic expression for the given phrase is \(20 - x\).

Step 1: Recognize referring number

The phrase refers to a number that we need to represent with a variable. In this case, we are told to let this number be represented by \(x\).

Step 2: Understand the operation

The phrase says 'A number subtracted from 20'. In algebraic terms, this represents the operation of subtraction where '20' is the starting value and 'x' is the quantity to be subtracted from it.

Step 3: Translate to an algebraic expression

Now that we understand the operation, we can translate the phrase into an algebraic expression. 'A number subtracted from 20' translates to '20 - x'.

Key Concepts

Algebraic Expressions

Once we begin to explore the world of algebra, we quickly encounter the concept of algebraic expressions. An expression in algebra is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that represents a particular value. Unlike equations, algebraic expressions don't have an equal sign; they are simply a way of representing a mathematical thought or relationship.

For example, the expression '2x + 3' tells us that some number 'x' is being doubled and then increased by 3. In the context of our exercise, converting an English phrase like 'A number subtracted from 20' into an algebraic expression requires identifying the variable and the operation involved. Ultimately, expressions offer a shorthand for conveying mathematical ideas quickly and succinctly, which is a fundamental skill in algebra.

Subtraction in Algebra

Subtraction is one of the core operations in algebra, and understanding how it works within expressions is critical. When we read a phrase like 'A number subtracted from 20,' it might sound a little backwards at first, because we're used to seeing '20 subtracted from a number.' However, in algebra, we write this as '20 - x'.

Order Matters

Unlike addition, the order in subtraction matters significantly. '20 - x' and 'x - 20' represent fundamentally different situations. The former indicates that whatever value 'x' has, it is being taken away from 20. This concept is sometimes referred to as 'difference' when discussing algebraic operations.

Understanding the rules governing subtraction is key to mastering algebra, as it not only affects how we write expressions but also how we solve them.

Variable Representation

Variables are the alphabet of algebra, serving as placeholders for values that we may not know yet. They allow us to write general formulas and expressions that can apply to many different situations. In the example given of 'A number subtracted from 20,' the variable used is 'x'.

It's important to realize that variables can represent anything: a specific but unknown quantity, a general value we can apply in various situations, or even a whole range of possible values. Choosing a variable is like picking a character for a role in a play; it needs to be consistent throughout your work on that problem and make sense within the context of the problem you're solving.

Being able to choose and use variables thoughtfully is a fundamental skill in algebra. As we represent English phrases as algebraic expressions, we are translating words into a symbolic language, where variables play a central role in the communication of mathematical ideas.