Problem 38 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. three less than the product of 13 and a number

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(13x - 3\)

Step 1: Identification of variables

According to the problem, \(x\) represents the number. Therefore, 'a number' in the phrase will be represented by the variable \(x\).

Step 2: Understanding 'product of'

In mathematics, 'product of' refers to the result obtained from multiplying numbers together. In the phrase 'product of 13 and a number', we will be multiplying 13 by our variable \(x\). This can be represented as \(13x\).

Step 3: Understanding 'less than'

The word 'less than' in mathematics signifies subtraction. In the phrase 'three less than the product of 13 and a number', it would mean three units are to be subtracted from the product of 13 and our variable \(x\). This can be written as \(13x - 3\)

Key Concepts

Mathematical Operations

Mathematical operations are the fundamental building blocks in algebra and mathematics as a whole. These operations include addition, subtraction, multiplication, and division. Each of these serves a specific purpose:

• Addition: Combining two or more quantities.
• Subtraction: Finding the difference between numbers by taking one away from another.
• Multiplication: A shorthand for repeated addition; finding the total of one number multiplied by another.
• Division: Splitting a number into equal parts or groups.

Understanding these operations is essential when working with algebraic expressions, as they help translate words into mathematical terms. For example, in the phrase 'product of 13 and a number', the term 'product' indicates that we need to use multiplication.

Variables in Algebra

Variables are symbols used to represent unknown values in algebra. They are commonly denoted by letters such as \(x\), \(y\), and \(z\). A variable can stand in for a number, and it gives us the flexibility to solve equations with unknowns.

In algebraic expressions, variables play a crucial role. For instance, when you see the phrase 'a number', it often hints that you should use a variable like \(x\). By using variables, we can generalize problems, making it easier to find solutions to similar scenarios without repeatedly solving the same problem.

In the context of our exercise, the variable \(x\) represents an unknown number. The presence of a variable allows us to write expressions such as \(13x\), where 13 and \(x\) are multiplied to form a product.

Translating Phrases to Expressions

Translating between verbal phrases and algebraic expressions is a key skill in mastering algebra. This involves converting the descriptive language of a phrase into mathematical symbols and operations. Such translations typically follow these guidelines:

• 'Product of': Use multiplication; e.g. 'product of 13 and a number' becomes \(13x\).
• 'Less than': Indicates subtraction from a subsequent quantity; e.g. 'three less than' means you subtract 3 from the result of the prior operation.

These translations demand a good grasp of both language and math. In our example, the phrase 'three less than the product of 13 and a number' is transformed into the expression \(13x - 3\). By learning which words correspond to mathematical operations, translating between language and math becomes a straightforward process.