Problem 38 - Introductory and Intermediate Algebra for College Students 4th

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 of 'three less than the product of 13 and a number' is \(13x - 3\).

Step 1: Identifying the Variable

The problem gives the hint that the variable we should use is \(x\). In this case, \(x\) will be representing the number in the phrase 'the product of 13 and a number'.

Step 2: Decoding 'Product'

In algebra, 'product' means multiplication. So the 'product of 13 and a number' translates to '13 multiplied by a number', this number is our variable \(x\), so our expression becomes \(13x\).

Step 3: Decoding 'Three Less Than'

The phrase 'three less than' is translated as 'subtract three from'. We are taking three less than the product of 13 and a number. Therefore, we subtract 3 from the result of the multiplication expression we got from step 2. Our final algebraic expression is \(13x - 3\).

Key Concepts

Mathematical Operations

Understanding mathematical operations is essential when working with algebraic expressions. Operations such as addition, subtraction, multiplication, and division form the basis of all algebraic manipulation.
In the context of our exercise, we encounter two key operations:

• Multiplication: The concept of 'product' in mathematics refers to the result of multiplying two numbers. Here, the product involves multiplying the number 13 by the variable, resulting in the term \(13x\).
• Subtraction: The phrase 'three less than' indicates a subtraction operation. This signifies removing 3 from the value obtained by the preceding multiplication operation, which gives us the expression \(13x - 3\).

Recognizing these operations in word problems is crucial as it helps in correctly forming the intended algebraic expressions.

Variables in Algebra

Variables are symbols, often letters, that represent unknown numbers in algebra. They are essential because they allow us to write general expressions that can apply to many different situations. In algebra, we usually work with letters like \(x\), \(y\), and \(z\) to denote these unknowns.
In this exercise, the variable \(x\) symbolizes the unknown number mentioned in the phrase. By assigning a variable, we can easily set up an expression or equation to solve various problems. The versatility of variables makes algebra a powerful tool for modeling real-world situations where values are not fixed or known.
By understanding how variables function, we gain the ability to translate complex worded problems into dynamic expressions and equations that become easier to manage and solve.

Translating Verbal Phrases to Algebraic Expressions

Translating verbal phrases into algebraic expressions is a pivotal skill in algebra. It involves interpreting everyday language and rephrasing it mathematically using symbols and numbers.
Let's break down this translation process using the exercise as an example:

• Identify key phrases: Understand words like 'product' and 'less than,' which imply specific mathematical operations.
• Assign a variable: Decipher which part of the phrase includes an unknown quantity that needs representation, in this case, by \(x\).
• Formulate the expression: Combine the understood operations and variable in a coherent algebraic expression, going step-by-step. From 'the product of 13 and a number' leading to \(13x\), to incorporating the subtraction 'three less than' to end with \(13x - 3\).

This method of translation allows us to turn complex narrative descriptions into simpler, more manageable algebraic expressions. Mastering this skill streamlines problem-solving and enhances comprehension of mathematical operations.