Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. nine decreased by a number

Step-by-Step Solution

Answer

The algebraic expression for 'nine decreased by a number' is \(9 - x\).

Step 1: Identify the Mathematical Operation

In the given phrase, 'nine decreased by a number', the term 'decreased by' corresponds to the mathematical operation of subtraction.

Step 2: Identify the Order of the Terms

'Nine decreased by a number' means we start with nine or \(9\), which is decreased by \(x\), i.e., the number. So, \(x\) is subtracted from \(9\).

Step 3: Write the Algebraic Expression

Based on the order of the terms found in previous step and the operation identified, the algebraic expression becomes \(9 - x\).

Key Concepts

Translating English Phrases to Algebra

When it comes to solving algebra problems, one of the first skills you need to master is translating English phrases into algebraic expressions. This is a crucial step, as it lays the foundation for solving mathematical problems that are described in words. To successfully translate a phrase, you must identify keywords and understand the operations they represent.

For example, words such as 'decreased by', 'less than', or 'subtract' all signal the need for a subtraction operation in your algebraic expression. Moreover, it's essential to pay attention to the order in which the terms are presented in the phrase. In the context of subtraction, the order determines which number is to be subtracted from which.

Identifying Operations and Terms

When faced with a phrase like 'nine decreased by a number', your approach should include:

• Identifying the operation: Here, 'decreased by' indicates subtraction.
• Understanding the terms: 'Nine' is the number we start with, and 'a number' represents the quantity to be subtracted, which we can call as variable 'x'.
• Translating the phrase: The phrase translates into the algebraic expression '9 - x', where 9 is decreased by the variable 'x'.

Being meticulous with this process ensures a correct translation from English phrases to algebraic expressions, improving your problem-solving skills.

Subtraction in Algebra

Subtraction is one of the fundamental operations in algebra, often encountered when manipulating algebraic expressions. The order in subtraction is particularly important and unlike addition, is not commutative; in other words, changing the order of the terms changes the result.

In algebra, when subtracting variables or numbers, the term being subtracted is taken away from the first term. If we have an expression like '9 - x', it is interpreted as 'start with 9 and then remove the amount represented by x'. This is different from 'x - 9', which would mean 'start with the amount represented by x and then remove 9'.

Subtraction Syntax in Algebra

An important aspect to remember is the syntax of subtraction in algebra:

• The minus sign (-) is used to indicate subtraction.
• The term directly after the minus sign is the one being subtracted.
• The expression altogether represents the difference between the first term and the second term.

Understanding and applying the correct order of terms is key to correctly interpreting and solving problems that involve subtraction in algebra.

Variable Representation

Variables are the placeholders or symbols in algebra that represent unknown values. Often denoted by letters such as 'x', 'y', or 'z', variables allow us to create general expressions that can be used to solve a wide variety of problems.

Choosing appropriate variable representation is an integral part of solving algebraic problems. In our example, 'a number' is a vague term that can represent any numerical value, so we assign it a variable, most commonly 'x'. This practice provides a concrete element to manipulate within the expression.

Using Variables Effectively

Effective use of variables involves:

• Selecting a variable to represent the unknown quantity.
• Maintaining consistency by using the same variable throughout a problem, unless there's more than one unknown.
• Understanding that the variable is simply a symbol that can be replaced with different numbers depending on the scenario.

By mastering variable representation, you increase your ability to communicate and solve mathematical expressions and equations with clarity and precision.